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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000668
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [3,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => [3,2,1]
=> [2,1]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 12%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => ([],3)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => ([(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([],6)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [1,6,5,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [6,5,1,4,2,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [5,6,1,4,2,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [6,1,5,4,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [1,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [1,5,6,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [5,4,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => [1,6,4,5,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [1,5,4,6,2,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [4,5,3,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [5,3,4,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [6,5,1,2,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,1,2] => [5,6,1,2,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,1,2] => [6,1,5,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,1,2] => [1,6,5,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,1,2] => [6,5,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,1,2] => [5,6,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,1,2] => [6,4,5,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,1,2] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => [6,1,2,5,4,3] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => [1,6,2,5,4,3] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,1,3] => [6,5,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,1,3] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,1,4] => [6,1,5,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,2,4] => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,7,2,3,4,5,6] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [6,7,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,2,7,3,4,5,6] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [1,7,6,2,3,4,5] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [6,7,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,6,7,2,3,4,5] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [1,2,3,7,4,5,6] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000375
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,2,1,3] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,2,3,6,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,2,6,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,2,6,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,2,5,6,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,6,2,4,5,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,5,2,6,4,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,6,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,4,6,2,5,3] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,5,6,3,2,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,1,3,6,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,1,6,3,5,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [5,1,6,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,1,5,6,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,1,4,5,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,1,6,4,3] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [5,2,6,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,1,4,5,2] => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,5,1,6,4,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,6,2,1,5,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,6,2,4,1,3] => 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,5,2,6,1,3] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,5,6,3,1,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [1,2,3,4,7,5,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [1,2,3,7,4,6,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [1,2,3,7,5,4,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [1,2,3,6,7,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [1,2,7,3,5,6,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => [1,2,6,3,7,5,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [1,2,7,4,3,6,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => [1,2,7,4,5,3,6] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [1,2,6,4,7,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [1,2,5,7,3,6,4] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => [1,2,6,7,5,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [1,2,6,7,4,3,5] => 1 = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [1,2,5,6,7,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [1,7,2,4,5,6,3] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [1,6,2,4,7,5,3] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [1,5,2,7,4,6,3] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [1,6,2,7,5,4,3] => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [1,5,2,6,7,4,3] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,3,2,5,6,4] => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [1,6,3,2,7,5,4] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [1,7,3,4,2,6,5] => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [1,5,3,7,2,6,4] => ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [1,6,3,7,4,2,5] => ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [1,4,7,2,5,6,3] => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [1,4,6,2,7,5,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [1,5,7,4,2,6,3] => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [1,5,7,3,2,6,4] => ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [1,6,7,3,5,2,4] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [1,5,6,3,7,2,4] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [1,4,5,7,2,6,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [1,5,6,7,4,2,3] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [7,1,3,4,5,6,2] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [6,1,3,4,7,5,2] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [5,1,3,7,4,6,2] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [6,1,3,7,5,4,2] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [5,1,3,6,7,4,2] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [4,1,7,3,5,6,2] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [4,1,6,3,7,5,2] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [5,1,7,4,3,6,2] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [6,1,7,4,5,3,2] => ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [5,1,6,4,7,3,2] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [4,1,5,7,3,6,2] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [4,1,6,7,5,3,2] => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,1,6,7,4,3,2] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [4,1,5,6,7,3,2] => ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [7,2,1,4,5,6,3] => ? = 1 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [6,2,1,4,7,5,3] => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [5,2,1,7,4,6,3] => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => [6,2,1,7,5,4,3] => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [5,2,1,6,7,4,3] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => [7,2,3,1,5,6,4] => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [6,2,3,1,7,5,4] => ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [6,2,3,7,4,1,5] => ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [4,2,7,1,5,6,3] => ? = 1 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [4,2,6,1,7,5,3] => ? = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [5,2,7,3,1,6,4] => ? = 2 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [6,2,7,3,5,1,4] => ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [5,2,6,3,7,1,4] => ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2,6,7,4,1,3] => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [3,7,1,4,5,6,2] => ? = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [3,6,1,4,7,5,2] => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [3,5,1,7,4,6,2] => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [3,6,1,7,5,4,2] => ? = 3 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [3,5,1,6,7,4,2] => ? = 3 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [4,7,3,1,5,6,2] => ? = 1 - 1
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000298
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000298: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000298: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ? = 2
[1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,2,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,3,4],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
=> ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,6),(0,7),(0,8),(1,4),(1,16),(2,5),(2,15),(3,2),(3,14),(4,10),(4,11),(5,9),(5,11),(6,12),(6,13),(7,12),(7,17),(8,3),(8,13),(8,17),(9,18),(10,18),(11,18),(12,19),(13,1),(13,19),(14,15),(15,9),(16,10),(17,14),(17,19),(19,16)],20)
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,2),(0,3),(0,10),(1,9),(1,11),(1,31),(2,1),(2,24),(3,6),(3,8),(3,30),(4,18),(4,27),(5,19),(5,20),(6,17),(6,29),(7,15),(7,21),(8,12),(8,26),(8,29),(9,16),(9,28),(10,24),(10,30),(11,14),(11,23),(11,28),(12,13),(12,22),(12,23),(13,32),(13,33),(14,33),(14,39),(15,37),(16,39),(17,38),(18,36),(19,34),(20,34),(21,5),(21,37),(22,21),(22,32),(23,25),(23,33),(24,31),(25,27),(25,35),(26,13),(26,38),(27,19),(27,36),(28,4),(28,25),(28,39),(29,7),(29,22),(29,38),(30,17),(30,26),(31,14),(31,16),(32,37),(33,35),(35,36),(36,34),(37,20),(38,15),(38,32),(39,18),(39,35)],40)
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,16),(0,17),(0,28),(1,23),(1,58),(2,19),(2,53),(3,11),(3,21),(3,80),(4,10),(4,20),(4,79),(5,12),(5,13),(5,81),(6,48),(6,73),(7,49),(7,74),(8,24),(8,75),(8,86),(9,25),(9,76),(9,87),(10,14),(10,71),(10,77),(11,15),(11,72),(11,78),(12,26),(12,69),(12,88),(13,27),(13,70),(13,88),(14,31),(14,82),(15,32),(15,83),(16,4),(16,36),(16,54),(17,3),(17,36),(17,55),(18,44),(18,67),(18,68),(19,39),(19,40),(19,43),(20,34),(20,56),(20,77),(21,35),(21,57),(21,78),(22,45),(22,46),(22,47),(23,18),(23,50),(23,82),(23,83),(24,41),(24,61),(24,63),(25,42),(25,62),(25,64),(26,37),(26,56),(26,84),(27,38),(27,57),(27,85),(28,54),(28,55),(28,81),(29,90),(29,118),(30,89),(30,117),(31,93),(32,94),(33,116),(33,124),(34,91),(34,122),(35,92),(35,123),(36,1),(36,121),(37,91),(37,110),(38,92),(38,111),(39,90),(39,104),(40,90),(40,105),(41,98),(41,108),(42,99),(42,109),(43,104),(43,105),(44,106),(44,107),(45,102),(45,103),(46,6),(46,89),(46,102),(47,7),(47,89),(47,103),(48,100),(49,101),(50,44),(50,93),(50,94),(51,39),(51,119),(52,40),(52,120),(53,43),(54,79),(54,121),(55,80),(55,121),(56,59),(56,91),(57,60),(57,92),(58,50),(59,86),(59,112),(60,87),(60,113),(61,65),(61,108),(62,66),(62,109),(63,51),(63,98),(64,52),(64,99),(65,29),(65,96),(66,29),(66,97),(67,61),(67,106),(67,124),(68,62),(68,107),(68,124),(69,37),(69,95),(70,38),(70,95),(71,31),(71,122),(72,32),(72,123),(73,51),(73,100),(74,52),(74,101),(75,41),(75,114),(76,42),(76,115),(77,8),(77,59),(77,122),(78,9),(78,60),(78,123),(79,34),(79,71),(80,35),(80,72),(81,69),(81,70),(82,33),(82,67),(82,93),(83,33),(83,68),(83,94),(84,30),(84,46),(84,110),(85,30),(85,47),(85,111),(86,63),(86,73),(86,114),(87,64),(87,74),(87,115),(88,22),(88,84),(88,85),(88,95),(89,2),(89,127),(90,125),(91,112),(92,113),(93,106),(93,116),(94,107),(94,116),(95,45),(95,110),(95,111),(96,118),(97,118),(98,119),(99,120),(100,119),(101,120),(102,48),(102,127),(103,49),(103,127),(104,125),(105,125),(106,108),(106,126),(107,109),(107,126),(108,96),(109,97),(110,102),(110,117),(111,103),(111,117),(112,114),(113,115),(114,98),(114,100),(115,99),(115,101),(116,126),(117,127),(118,125),(119,104),(120,105),(121,58),(122,75),(122,112),(123,76),(123,113),(124,65),(124,66),(124,126),(126,96),(126,97),(127,53)],128)
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,4),(0,8),(1,7),(1,21),(2,6),(2,20),(3,2),(3,18),(4,1),(4,19),(5,12),(5,13),(6,11),(6,16),(7,11),(7,17),(8,5),(8,18),(8,19),(9,23),(10,23),(11,22),(12,14),(13,15),(14,9),(15,10),(16,9),(16,22),(17,10),(17,22),(18,12),(18,20),(19,13),(19,21),(20,14),(20,16),(21,15),(21,17),(22,23)],24)
=> ? = 3
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,2),(0,3),(1,8),(1,12),(2,1),(3,4),(3,7),(4,25),(5,17),(6,24),(7,13),(7,25),(8,10),(8,28),(9,27),(10,9),(10,29),(11,22),(11,23),(12,21),(12,28),(13,18),(13,21),(14,30),(15,30),(17,6),(18,17),(19,16),(20,15),(20,31),(21,26),(22,24),(22,31),(23,14),(23,31),(24,19),(25,5),(25,18),(26,20),(26,22),(27,14),(27,15),(28,11),(28,26),(28,29),(29,20),(29,23),(29,27),(30,16),(31,19),(31,30)],32)
=> ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,9),(0,10),(1,2),(2,7),(2,11),(3,23),(4,17),(5,22),(6,13),(6,24),(7,8),(7,27),(8,26),(9,1),(9,21),(10,3),(10,21),(11,19),(11,27),(12,15),(12,20),(13,18),(13,19),(14,28),(15,28),(17,5),(18,17),(19,25),(20,22),(20,28),(21,6),(21,23),(22,16),(23,24),(24,4),(24,18),(25,14),(25,20),(26,14),(26,15),(27,12),(27,25),(27,26),(28,16)],29)
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,4),(0,14),(0,15),(1,26),(2,6),(2,27),(3,13),(3,28),(4,11),(4,12),(5,112),(6,5),(6,96),(7,32),(7,111),(8,23),(8,122),(9,24),(9,109),(10,25),(10,108),(11,22),(11,123),(12,21),(12,34),(12,123),(13,17),(13,20),(13,119),(14,2),(14,48),(15,3),(15,48),(16,93),(16,95),(17,120),(17,126),(18,90),(18,113),(19,35),(19,115),(19,118),(20,29),(20,120),(20,124),(21,94),(21,117),(22,97),(22,125),(23,89),(23,116),(24,87),(24,91),(25,88),(25,92),(26,30),(26,112),(26,126),(27,96),(27,114),(28,97),(28,119),(29,66),(29,106),(30,63),(30,99),(31,45),(31,79),(31,82),(32,44),(32,80),(32,81),(33,47),(33,57),(33,86),(34,94),(34,105),(34,114),(35,65),(35,71),(35,103),(36,131),(37,133),(38,130),(38,132),(39,132),(39,166),(40,133),(40,135),(41,127),(42,136),(42,143),(42,144),(43,129),(43,137),(44,129),(44,138),(45,134),(45,141),(46,10),(47,9),(47,166),(48,1),(49,153),(50,134),(50,135),(50,149),(51,139),(52,140),(52,159),(53,155),(54,143),(54,148),(55,144),(55,147),(56,150),(57,18),(57,130),(57,166),(58,44),(58,127),(59,76),(59,165),(60,80),(60,164),(61,70),(62,88),(62,139),(63,89),(63,165),(64,62),(65,60),(65,128),(66,107),(67,49),(68,53),(68,137),(69,51),(70,75),(71,73),(71,163),(72,108),(72,159),(73,43),(73,161),(74,42),(74,142),(75,41),(75,160),(76,37),(76,157),(77,36),(77,162),(78,40),(78,157),(78,163),(79,72),(79,141),(79,167),(80,110),(80,129),(81,62),(81,138),(82,83),(82,134),(82,167),(83,84),(83,151),(84,55),(84,152),(85,95),(85,146),(86,16),(86,85),(86,130),(87,69),(88,56),(88,136),(89,104),(90,46),(91,64),(92,49),(92,136),(93,58),(93,160),(94,61),(94,158),(95,87),(95,160),(96,8),(96,101),(97,102),(98,45),(98,50),(98,131),(99,71),(99,78),(99,165),(100,52),(100,72),(100,164),(101,70),(101,122),(102,77),(102,115),(103,50),(103,82),(103,128),(103,163),(104,43),(104,68),(105,38),(105,86),(105,158),(106,36),(106,98),(107,37),(107,40),(108,67),(108,92),(109,46),(109,91),(110,42),(110,54),(110,154),(111,64),(111,81),(112,59),(112,63),(113,60),(113,100),(114,61),(114,101),(115,65),(115,113),(115,162),(116,41),(116,58),(117,39),(117,47),(117,158),(118,31),(118,98),(118,103),(118,162),(119,19),(119,102),(119,124),(120,66),(120,121),(121,76),(121,78),(121,107),(122,75),(122,93),(122,116),(123,33),(123,105),(123,117),(123,125),(124,77),(124,106),(124,118),(125,38),(125,39),(125,57),(126,59),(126,99),(126,121),(127,51),(127,138),(128,149),(128,164),(128,167),(129,154),(130,7),(130,146),(131,52),(131,141),(131,149),(132,146),(133,53),(133,156),(134,151),(134,168),(135,151),(135,156),(136,150),(136,153),(137,55),(137,154),(137,155),(138,139),(139,56),(140,54),(140,142),(141,159),(141,168),(142,143),(143,153),(143,169),(144,150),(144,169),(146,111),(147,169),(148,169),(149,140),(149,168),(150,145),(151,152),(152,147),(153,145),(154,144),(154,148),(155,147),(155,148),(156,152),(156,155),(157,68),(157,133),(157,161),(158,85),(158,132),(159,67),(160,69),(160,127),(161,84),(161,137),(161,156),(162,79),(162,100),(162,128),(162,131),(163,83),(163,135),(163,161),(164,74),(164,110),(164,140),(165,73),(165,104),(165,157),(166,90),(166,109),(167,74),(167,168),(168,142),(169,145)],170)
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,13),(0,14),(0,30),(1,28),(2,95),(3,94),(4,77),(5,10),(5,110),(6,70),(7,8),(7,31),(7,71),(8,12),(8,99),(9,23),(9,100),(10,22),(10,111),(11,20),(11,98),(12,17),(12,18),(12,112),(13,7),(13,64),(14,5),(14,27),(14,63),(15,53),(15,87),(16,82),(16,104),(17,55),(17,107),(18,55),(18,105),(19,52),(19,80),(20,78),(20,84),(21,79),(21,86),(22,54),(22,108),(23,54),(23,106),(24,51),(24,83),(25,29),(25,102),(25,103),(26,33),(26,85),(26,109),(27,32),(27,101),(27,110),(28,50),(28,81),(29,62),(29,88),(30,63),(30,64),(31,65),(31,68),(31,99),(32,68),(32,69),(32,93),(33,36),(33,42),(33,76),(34,118),(34,134),(35,121),(35,142),(36,118),(36,141),(37,3),(37,117),(38,116),(38,126),(39,115),(39,126),(40,121),(40,129),(41,114),(41,125),(42,118),(42,127),(43,117),(44,133),(45,132),(46,123),(47,128),(48,124),(49,125),(50,114),(51,6),(51,139),(52,2),(52,115),(53,16),(53,138),(54,119),(55,136),(56,84),(56,140),(57,60),(58,73),(58,135),(59,47),(59,140),(60,50),(61,51),(61,116),(62,56),(62,113),(63,9),(63,101),(64,71),(65,77),(65,137),(66,102),(67,38),(67,138),(68,89),(68,137),(69,90),(69,137),(70,57),(71,4),(71,65),(72,85),(72,129),(73,81),(73,123),(74,34),(74,122),(75,41),(75,130),(76,58),(76,127),(76,141),(77,25),(77,66),(78,60),(79,82),(79,120),(80,45),(80,115),(81,44),(81,114),(82,97),(83,37),(83,139),(84,91),(85,42),(85,131),(86,26),(86,72),(86,120),(87,24),(87,61),(87,138),(88,34),(88,36),(88,113),(89,79),(89,92),(90,39),(90,52),(91,41),(91,49),(92,35),(92,40),(92,120),(93,53),(93,67),(94,46),(94,73),(95,56),(95,59),(96,58),(96,94),(96,124),(97,48),(97,96),(98,57),(98,78),(99,21),(99,89),(99,112),(100,19),(100,90),(100,106),(101,69),(101,100),(102,62),(102,95),(102,142),(103,74),(103,88),(103,142),(104,37),(104,43),(105,35),(105,103),(105,136),(106,39),(106,80),(106,119),(107,40),(107,72),(107,136),(108,38),(108,61),(108,119),(109,76),(109,96),(109,131),(110,15),(110,93),(110,111),(111,67),(111,87),(111,108),(112,86),(112,92),(112,105),(112,107),(113,134),(113,140),(113,141),(114,133),(115,11),(115,132),(116,43),(116,139),(117,1),(118,143),(119,45),(119,126),(120,97),(120,109),(120,129),(121,122),(122,47),(122,134),(123,44),(124,46),(124,135),(125,133),(126,132),(127,135),(127,143),(128,49),(128,130),(129,48),(129,131),(130,125),(131,124),(131,127),(132,98),(134,128),(134,143),(135,123),(136,74),(136,121),(137,66),(138,83),(138,104),(138,116),(139,70),(139,117),(140,75),(140,91),(140,128),(141,75),(141,143),(142,59),(142,113),(142,122),(143,130)],144)
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000307
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000307: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000307: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ? = 2
[1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,2,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,3,4],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
=> ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,6),(0,7),(0,8),(1,4),(1,16),(2,5),(2,15),(3,2),(3,14),(4,10),(4,11),(5,9),(5,11),(6,12),(6,13),(7,12),(7,17),(8,3),(8,13),(8,17),(9,18),(10,18),(11,18),(12,19),(13,1),(13,19),(14,15),(15,9),(16,10),(17,14),(17,19),(19,16)],20)
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,2),(0,3),(0,10),(1,9),(1,11),(1,31),(2,1),(2,24),(3,6),(3,8),(3,30),(4,18),(4,27),(5,19),(5,20),(6,17),(6,29),(7,15),(7,21),(8,12),(8,26),(8,29),(9,16),(9,28),(10,24),(10,30),(11,14),(11,23),(11,28),(12,13),(12,22),(12,23),(13,32),(13,33),(14,33),(14,39),(15,37),(16,39),(17,38),(18,36),(19,34),(20,34),(21,5),(21,37),(22,21),(22,32),(23,25),(23,33),(24,31),(25,27),(25,35),(26,13),(26,38),(27,19),(27,36),(28,4),(28,25),(28,39),(29,7),(29,22),(29,38),(30,17),(30,26),(31,14),(31,16),(32,37),(33,35),(35,36),(36,34),(37,20),(38,15),(38,32),(39,18),(39,35)],40)
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,16),(0,17),(0,28),(1,23),(1,58),(2,19),(2,53),(3,11),(3,21),(3,80),(4,10),(4,20),(4,79),(5,12),(5,13),(5,81),(6,48),(6,73),(7,49),(7,74),(8,24),(8,75),(8,86),(9,25),(9,76),(9,87),(10,14),(10,71),(10,77),(11,15),(11,72),(11,78),(12,26),(12,69),(12,88),(13,27),(13,70),(13,88),(14,31),(14,82),(15,32),(15,83),(16,4),(16,36),(16,54),(17,3),(17,36),(17,55),(18,44),(18,67),(18,68),(19,39),(19,40),(19,43),(20,34),(20,56),(20,77),(21,35),(21,57),(21,78),(22,45),(22,46),(22,47),(23,18),(23,50),(23,82),(23,83),(24,41),(24,61),(24,63),(25,42),(25,62),(25,64),(26,37),(26,56),(26,84),(27,38),(27,57),(27,85),(28,54),(28,55),(28,81),(29,90),(29,118),(30,89),(30,117),(31,93),(32,94),(33,116),(33,124),(34,91),(34,122),(35,92),(35,123),(36,1),(36,121),(37,91),(37,110),(38,92),(38,111),(39,90),(39,104),(40,90),(40,105),(41,98),(41,108),(42,99),(42,109),(43,104),(43,105),(44,106),(44,107),(45,102),(45,103),(46,6),(46,89),(46,102),(47,7),(47,89),(47,103),(48,100),(49,101),(50,44),(50,93),(50,94),(51,39),(51,119),(52,40),(52,120),(53,43),(54,79),(54,121),(55,80),(55,121),(56,59),(56,91),(57,60),(57,92),(58,50),(59,86),(59,112),(60,87),(60,113),(61,65),(61,108),(62,66),(62,109),(63,51),(63,98),(64,52),(64,99),(65,29),(65,96),(66,29),(66,97),(67,61),(67,106),(67,124),(68,62),(68,107),(68,124),(69,37),(69,95),(70,38),(70,95),(71,31),(71,122),(72,32),(72,123),(73,51),(73,100),(74,52),(74,101),(75,41),(75,114),(76,42),(76,115),(77,8),(77,59),(77,122),(78,9),(78,60),(78,123),(79,34),(79,71),(80,35),(80,72),(81,69),(81,70),(82,33),(82,67),(82,93),(83,33),(83,68),(83,94),(84,30),(84,46),(84,110),(85,30),(85,47),(85,111),(86,63),(86,73),(86,114),(87,64),(87,74),(87,115),(88,22),(88,84),(88,85),(88,95),(89,2),(89,127),(90,125),(91,112),(92,113),(93,106),(93,116),(94,107),(94,116),(95,45),(95,110),(95,111),(96,118),(97,118),(98,119),(99,120),(100,119),(101,120),(102,48),(102,127),(103,49),(103,127),(104,125),(105,125),(106,108),(106,126),(107,109),(107,126),(108,96),(109,97),(110,102),(110,117),(111,103),(111,117),(112,114),(113,115),(114,98),(114,100),(115,99),(115,101),(116,126),(117,127),(118,125),(119,104),(120,105),(121,58),(122,75),(122,112),(123,76),(123,113),(124,65),(124,66),(124,126),(126,96),(126,97),(127,53)],128)
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,4),(0,8),(1,7),(1,21),(2,6),(2,20),(3,2),(3,18),(4,1),(4,19),(5,12),(5,13),(6,11),(6,16),(7,11),(7,17),(8,5),(8,18),(8,19),(9,23),(10,23),(11,22),(12,14),(13,15),(14,9),(15,10),(16,9),(16,22),(17,10),(17,22),(18,12),(18,20),(19,13),(19,21),(20,14),(20,16),(21,15),(21,17),(22,23)],24)
=> ? = 3
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,2),(0,3),(1,8),(1,12),(2,1),(3,4),(3,7),(4,25),(5,17),(6,24),(7,13),(7,25),(8,10),(8,28),(9,27),(10,9),(10,29),(11,22),(11,23),(12,21),(12,28),(13,18),(13,21),(14,30),(15,30),(17,6),(18,17),(19,16),(20,15),(20,31),(21,26),(22,24),(22,31),(23,14),(23,31),(24,19),(25,5),(25,18),(26,20),(26,22),(27,14),(27,15),(28,11),(28,26),(28,29),(29,20),(29,23),(29,27),(30,16),(31,19),(31,30)],32)
=> ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,9),(0,10),(1,2),(2,7),(2,11),(3,23),(4,17),(5,22),(6,13),(6,24),(7,8),(7,27),(8,26),(9,1),(9,21),(10,3),(10,21),(11,19),(11,27),(12,15),(12,20),(13,18),(13,19),(14,28),(15,28),(17,5),(18,17),(19,25),(20,22),(20,28),(21,6),(21,23),(22,16),(23,24),(24,4),(24,18),(25,14),(25,20),(26,14),(26,15),(27,12),(27,25),(27,26),(28,16)],29)
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,4),(0,14),(0,15),(1,26),(2,6),(2,27),(3,13),(3,28),(4,11),(4,12),(5,112),(6,5),(6,96),(7,32),(7,111),(8,23),(8,122),(9,24),(9,109),(10,25),(10,108),(11,22),(11,123),(12,21),(12,34),(12,123),(13,17),(13,20),(13,119),(14,2),(14,48),(15,3),(15,48),(16,93),(16,95),(17,120),(17,126),(18,90),(18,113),(19,35),(19,115),(19,118),(20,29),(20,120),(20,124),(21,94),(21,117),(22,97),(22,125),(23,89),(23,116),(24,87),(24,91),(25,88),(25,92),(26,30),(26,112),(26,126),(27,96),(27,114),(28,97),(28,119),(29,66),(29,106),(30,63),(30,99),(31,45),(31,79),(31,82),(32,44),(32,80),(32,81),(33,47),(33,57),(33,86),(34,94),(34,105),(34,114),(35,65),(35,71),(35,103),(36,131),(37,133),(38,130),(38,132),(39,132),(39,166),(40,133),(40,135),(41,127),(42,136),(42,143),(42,144),(43,129),(43,137),(44,129),(44,138),(45,134),(45,141),(46,10),(47,9),(47,166),(48,1),(49,153),(50,134),(50,135),(50,149),(51,139),(52,140),(52,159),(53,155),(54,143),(54,148),(55,144),(55,147),(56,150),(57,18),(57,130),(57,166),(58,44),(58,127),(59,76),(59,165),(60,80),(60,164),(61,70),(62,88),(62,139),(63,89),(63,165),(64,62),(65,60),(65,128),(66,107),(67,49),(68,53),(68,137),(69,51),(70,75),(71,73),(71,163),(72,108),(72,159),(73,43),(73,161),(74,42),(74,142),(75,41),(75,160),(76,37),(76,157),(77,36),(77,162),(78,40),(78,157),(78,163),(79,72),(79,141),(79,167),(80,110),(80,129),(81,62),(81,138),(82,83),(82,134),(82,167),(83,84),(83,151),(84,55),(84,152),(85,95),(85,146),(86,16),(86,85),(86,130),(87,69),(88,56),(88,136),(89,104),(90,46),(91,64),(92,49),(92,136),(93,58),(93,160),(94,61),(94,158),(95,87),(95,160),(96,8),(96,101),(97,102),(98,45),(98,50),(98,131),(99,71),(99,78),(99,165),(100,52),(100,72),(100,164),(101,70),(101,122),(102,77),(102,115),(103,50),(103,82),(103,128),(103,163),(104,43),(104,68),(105,38),(105,86),(105,158),(106,36),(106,98),(107,37),(107,40),(108,67),(108,92),(109,46),(109,91),(110,42),(110,54),(110,154),(111,64),(111,81),(112,59),(112,63),(113,60),(113,100),(114,61),(114,101),(115,65),(115,113),(115,162),(116,41),(116,58),(117,39),(117,47),(117,158),(118,31),(118,98),(118,103),(118,162),(119,19),(119,102),(119,124),(120,66),(120,121),(121,76),(121,78),(121,107),(122,75),(122,93),(122,116),(123,33),(123,105),(123,117),(123,125),(124,77),(124,106),(124,118),(125,38),(125,39),(125,57),(126,59),(126,99),(126,121),(127,51),(127,138),(128,149),(128,164),(128,167),(129,154),(130,7),(130,146),(131,52),(131,141),(131,149),(132,146),(133,53),(133,156),(134,151),(134,168),(135,151),(135,156),(136,150),(136,153),(137,55),(137,154),(137,155),(138,139),(139,56),(140,54),(140,142),(141,159),(141,168),(142,143),(143,153),(143,169),(144,150),(144,169),(146,111),(147,169),(148,169),(149,140),(149,168),(150,145),(151,152),(152,147),(153,145),(154,144),(154,148),(155,147),(155,148),(156,152),(156,155),(157,68),(157,133),(157,161),(158,85),(158,132),(159,67),(160,69),(160,127),(161,84),(161,137),(161,156),(162,79),(162,100),(162,128),(162,131),(163,83),(163,135),(163,161),(164,74),(164,110),(164,140),(165,73),(165,104),(165,157),(166,90),(166,109),(167,74),(167,168),(168,142),(169,145)],170)
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,13),(0,14),(0,30),(1,28),(2,95),(3,94),(4,77),(5,10),(5,110),(6,70),(7,8),(7,31),(7,71),(8,12),(8,99),(9,23),(9,100),(10,22),(10,111),(11,20),(11,98),(12,17),(12,18),(12,112),(13,7),(13,64),(14,5),(14,27),(14,63),(15,53),(15,87),(16,82),(16,104),(17,55),(17,107),(18,55),(18,105),(19,52),(19,80),(20,78),(20,84),(21,79),(21,86),(22,54),(22,108),(23,54),(23,106),(24,51),(24,83),(25,29),(25,102),(25,103),(26,33),(26,85),(26,109),(27,32),(27,101),(27,110),(28,50),(28,81),(29,62),(29,88),(30,63),(30,64),(31,65),(31,68),(31,99),(32,68),(32,69),(32,93),(33,36),(33,42),(33,76),(34,118),(34,134),(35,121),(35,142),(36,118),(36,141),(37,3),(37,117),(38,116),(38,126),(39,115),(39,126),(40,121),(40,129),(41,114),(41,125),(42,118),(42,127),(43,117),(44,133),(45,132),(46,123),(47,128),(48,124),(49,125),(50,114),(51,6),(51,139),(52,2),(52,115),(53,16),(53,138),(54,119),(55,136),(56,84),(56,140),(57,60),(58,73),(58,135),(59,47),(59,140),(60,50),(61,51),(61,116),(62,56),(62,113),(63,9),(63,101),(64,71),(65,77),(65,137),(66,102),(67,38),(67,138),(68,89),(68,137),(69,90),(69,137),(70,57),(71,4),(71,65),(72,85),(72,129),(73,81),(73,123),(74,34),(74,122),(75,41),(75,130),(76,58),(76,127),(76,141),(77,25),(77,66),(78,60),(79,82),(79,120),(80,45),(80,115),(81,44),(81,114),(82,97),(83,37),(83,139),(84,91),(85,42),(85,131),(86,26),(86,72),(86,120),(87,24),(87,61),(87,138),(88,34),(88,36),(88,113),(89,79),(89,92),(90,39),(90,52),(91,41),(91,49),(92,35),(92,40),(92,120),(93,53),(93,67),(94,46),(94,73),(95,56),(95,59),(96,58),(96,94),(96,124),(97,48),(97,96),(98,57),(98,78),(99,21),(99,89),(99,112),(100,19),(100,90),(100,106),(101,69),(101,100),(102,62),(102,95),(102,142),(103,74),(103,88),(103,142),(104,37),(104,43),(105,35),(105,103),(105,136),(106,39),(106,80),(106,119),(107,40),(107,72),(107,136),(108,38),(108,61),(108,119),(109,76),(109,96),(109,131),(110,15),(110,93),(110,111),(111,67),(111,87),(111,108),(112,86),(112,92),(112,105),(112,107),(113,134),(113,140),(113,141),(114,133),(115,11),(115,132),(116,43),(116,139),(117,1),(118,143),(119,45),(119,126),(120,97),(120,109),(120,129),(121,122),(122,47),(122,134),(123,44),(124,46),(124,135),(125,133),(126,132),(127,135),(127,143),(128,49),(128,130),(129,48),(129,131),(130,125),(131,124),(131,127),(132,98),(134,128),(134,143),(135,123),(136,74),(136,121),(137,66),(138,83),(138,104),(138,116),(139,70),(139,117),(140,75),(140,91),(140,128),(141,75),(141,143),(142,59),(142,113),(142,122),(143,130)],144)
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000632
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000632: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000632: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,2,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
=> ? = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,3,4],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,6),(0,7),(0,8),(1,4),(1,16),(2,5),(2,15),(3,2),(3,14),(4,10),(4,11),(5,9),(5,11),(6,12),(6,13),(7,12),(7,17),(8,3),(8,13),(8,17),(9,18),(10,18),(11,18),(12,19),(13,1),(13,19),(14,15),(15,9),(16,10),(17,14),(17,19),(19,16)],20)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,2),(0,3),(0,10),(1,9),(1,11),(1,31),(2,1),(2,24),(3,6),(3,8),(3,30),(4,18),(4,27),(5,19),(5,20),(6,17),(6,29),(7,15),(7,21),(8,12),(8,26),(8,29),(9,16),(9,28),(10,24),(10,30),(11,14),(11,23),(11,28),(12,13),(12,22),(12,23),(13,32),(13,33),(14,33),(14,39),(15,37),(16,39),(17,38),(18,36),(19,34),(20,34),(21,5),(21,37),(22,21),(22,32),(23,25),(23,33),(24,31),(25,27),(25,35),(26,13),(26,38),(27,19),(27,36),(28,4),(28,25),(28,39),(29,7),(29,22),(29,38),(30,17),(30,26),(31,14),(31,16),(32,37),(33,35),(35,36),(36,34),(37,20),(38,15),(38,32),(39,18),(39,35)],40)
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,16),(0,17),(0,28),(1,23),(1,58),(2,19),(2,53),(3,11),(3,21),(3,80),(4,10),(4,20),(4,79),(5,12),(5,13),(5,81),(6,48),(6,73),(7,49),(7,74),(8,24),(8,75),(8,86),(9,25),(9,76),(9,87),(10,14),(10,71),(10,77),(11,15),(11,72),(11,78),(12,26),(12,69),(12,88),(13,27),(13,70),(13,88),(14,31),(14,82),(15,32),(15,83),(16,4),(16,36),(16,54),(17,3),(17,36),(17,55),(18,44),(18,67),(18,68),(19,39),(19,40),(19,43),(20,34),(20,56),(20,77),(21,35),(21,57),(21,78),(22,45),(22,46),(22,47),(23,18),(23,50),(23,82),(23,83),(24,41),(24,61),(24,63),(25,42),(25,62),(25,64),(26,37),(26,56),(26,84),(27,38),(27,57),(27,85),(28,54),(28,55),(28,81),(29,90),(29,118),(30,89),(30,117),(31,93),(32,94),(33,116),(33,124),(34,91),(34,122),(35,92),(35,123),(36,1),(36,121),(37,91),(37,110),(38,92),(38,111),(39,90),(39,104),(40,90),(40,105),(41,98),(41,108),(42,99),(42,109),(43,104),(43,105),(44,106),(44,107),(45,102),(45,103),(46,6),(46,89),(46,102),(47,7),(47,89),(47,103),(48,100),(49,101),(50,44),(50,93),(50,94),(51,39),(51,119),(52,40),(52,120),(53,43),(54,79),(54,121),(55,80),(55,121),(56,59),(56,91),(57,60),(57,92),(58,50),(59,86),(59,112),(60,87),(60,113),(61,65),(61,108),(62,66),(62,109),(63,51),(63,98),(64,52),(64,99),(65,29),(65,96),(66,29),(66,97),(67,61),(67,106),(67,124),(68,62),(68,107),(68,124),(69,37),(69,95),(70,38),(70,95),(71,31),(71,122),(72,32),(72,123),(73,51),(73,100),(74,52),(74,101),(75,41),(75,114),(76,42),(76,115),(77,8),(77,59),(77,122),(78,9),(78,60),(78,123),(79,34),(79,71),(80,35),(80,72),(81,69),(81,70),(82,33),(82,67),(82,93),(83,33),(83,68),(83,94),(84,30),(84,46),(84,110),(85,30),(85,47),(85,111),(86,63),(86,73),(86,114),(87,64),(87,74),(87,115),(88,22),(88,84),(88,85),(88,95),(89,2),(89,127),(90,125),(91,112),(92,113),(93,106),(93,116),(94,107),(94,116),(95,45),(95,110),(95,111),(96,118),(97,118),(98,119),(99,120),(100,119),(101,120),(102,48),(102,127),(103,49),(103,127),(104,125),(105,125),(106,108),(106,126),(107,109),(107,126),(108,96),(109,97),(110,102),(110,117),(111,103),(111,117),(112,114),(113,115),(114,98),(114,100),(115,99),(115,101),(116,126),(117,127),(118,125),(119,104),(120,105),(121,58),(122,75),(122,112),(123,76),(123,113),(124,65),(124,66),(124,126),(126,96),(126,97),(127,53)],128)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,4),(0,8),(1,7),(1,21),(2,6),(2,20),(3,2),(3,18),(4,1),(4,19),(5,12),(5,13),(6,11),(6,16),(7,11),(7,17),(8,5),(8,18),(8,19),(9,23),(10,23),(11,22),(12,14),(13,15),(14,9),(15,10),(16,9),(16,22),(17,10),(17,22),(18,12),(18,20),(19,13),(19,21),(20,14),(20,16),(21,15),(21,17),(22,23)],24)
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,2),(0,3),(1,8),(1,12),(2,1),(3,4),(3,7),(4,25),(5,17),(6,24),(7,13),(7,25),(8,10),(8,28),(9,27),(10,9),(10,29),(11,22),(11,23),(12,21),(12,28),(13,18),(13,21),(14,30),(15,30),(17,6),(18,17),(19,16),(20,15),(20,31),(21,26),(22,24),(22,31),(23,14),(23,31),(24,19),(25,5),(25,18),(26,20),(26,22),(27,14),(27,15),(28,11),(28,26),(28,29),(29,20),(29,23),(29,27),(30,16),(31,19),(31,30)],32)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
=> ? = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,9),(0,10),(1,2),(2,7),(2,11),(3,23),(4,17),(5,22),(6,13),(6,24),(7,8),(7,27),(8,26),(9,1),(9,21),(10,3),(10,21),(11,19),(11,27),(12,15),(12,20),(13,18),(13,19),(14,28),(15,28),(17,5),(18,17),(19,25),(20,22),(20,28),(21,6),(21,23),(22,16),(23,24),(24,4),(24,18),(25,14),(25,20),(26,14),(26,15),(27,12),(27,25),(27,26),(28,16)],29)
=> ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,4),(0,14),(0,15),(1,26),(2,6),(2,27),(3,13),(3,28),(4,11),(4,12),(5,112),(6,5),(6,96),(7,32),(7,111),(8,23),(8,122),(9,24),(9,109),(10,25),(10,108),(11,22),(11,123),(12,21),(12,34),(12,123),(13,17),(13,20),(13,119),(14,2),(14,48),(15,3),(15,48),(16,93),(16,95),(17,120),(17,126),(18,90),(18,113),(19,35),(19,115),(19,118),(20,29),(20,120),(20,124),(21,94),(21,117),(22,97),(22,125),(23,89),(23,116),(24,87),(24,91),(25,88),(25,92),(26,30),(26,112),(26,126),(27,96),(27,114),(28,97),(28,119),(29,66),(29,106),(30,63),(30,99),(31,45),(31,79),(31,82),(32,44),(32,80),(32,81),(33,47),(33,57),(33,86),(34,94),(34,105),(34,114),(35,65),(35,71),(35,103),(36,131),(37,133),(38,130),(38,132),(39,132),(39,166),(40,133),(40,135),(41,127),(42,136),(42,143),(42,144),(43,129),(43,137),(44,129),(44,138),(45,134),(45,141),(46,10),(47,9),(47,166),(48,1),(49,153),(50,134),(50,135),(50,149),(51,139),(52,140),(52,159),(53,155),(54,143),(54,148),(55,144),(55,147),(56,150),(57,18),(57,130),(57,166),(58,44),(58,127),(59,76),(59,165),(60,80),(60,164),(61,70),(62,88),(62,139),(63,89),(63,165),(64,62),(65,60),(65,128),(66,107),(67,49),(68,53),(68,137),(69,51),(70,75),(71,73),(71,163),(72,108),(72,159),(73,43),(73,161),(74,42),(74,142),(75,41),(75,160),(76,37),(76,157),(77,36),(77,162),(78,40),(78,157),(78,163),(79,72),(79,141),(79,167),(80,110),(80,129),(81,62),(81,138),(82,83),(82,134),(82,167),(83,84),(83,151),(84,55),(84,152),(85,95),(85,146),(86,16),(86,85),(86,130),(87,69),(88,56),(88,136),(89,104),(90,46),(91,64),(92,49),(92,136),(93,58),(93,160),(94,61),(94,158),(95,87),(95,160),(96,8),(96,101),(97,102),(98,45),(98,50),(98,131),(99,71),(99,78),(99,165),(100,52),(100,72),(100,164),(101,70),(101,122),(102,77),(102,115),(103,50),(103,82),(103,128),(103,163),(104,43),(104,68),(105,38),(105,86),(105,158),(106,36),(106,98),(107,37),(107,40),(108,67),(108,92),(109,46),(109,91),(110,42),(110,54),(110,154),(111,64),(111,81),(112,59),(112,63),(113,60),(113,100),(114,61),(114,101),(115,65),(115,113),(115,162),(116,41),(116,58),(117,39),(117,47),(117,158),(118,31),(118,98),(118,103),(118,162),(119,19),(119,102),(119,124),(120,66),(120,121),(121,76),(121,78),(121,107),(122,75),(122,93),(122,116),(123,33),(123,105),(123,117),(123,125),(124,77),(124,106),(124,118),(125,38),(125,39),(125,57),(126,59),(126,99),(126,121),(127,51),(127,138),(128,149),(128,164),(128,167),(129,154),(130,7),(130,146),(131,52),(131,141),(131,149),(132,146),(133,53),(133,156),(134,151),(134,168),(135,151),(135,156),(136,150),(136,153),(137,55),(137,154),(137,155),(138,139),(139,56),(140,54),(140,142),(141,159),(141,168),(142,143),(143,153),(143,169),(144,150),(144,169),(146,111),(147,169),(148,169),(149,140),(149,168),(150,145),(151,152),(152,147),(153,145),(154,144),(154,148),(155,147),(155,148),(156,152),(156,155),(157,68),(157,133),(157,161),(158,85),(158,132),(159,67),(160,69),(160,127),(161,84),(161,137),(161,156),(162,79),(162,100),(162,128),(162,131),(163,83),(163,135),(163,161),(164,74),(164,110),(164,140),(165,73),(165,104),(165,157),(166,90),(166,109),(167,74),(167,168),(168,142),(169,145)],170)
=> ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,13),(0,14),(0,30),(1,28),(2,95),(3,94),(4,77),(5,10),(5,110),(6,70),(7,8),(7,31),(7,71),(8,12),(8,99),(9,23),(9,100),(10,22),(10,111),(11,20),(11,98),(12,17),(12,18),(12,112),(13,7),(13,64),(14,5),(14,27),(14,63),(15,53),(15,87),(16,82),(16,104),(17,55),(17,107),(18,55),(18,105),(19,52),(19,80),(20,78),(20,84),(21,79),(21,86),(22,54),(22,108),(23,54),(23,106),(24,51),(24,83),(25,29),(25,102),(25,103),(26,33),(26,85),(26,109),(27,32),(27,101),(27,110),(28,50),(28,81),(29,62),(29,88),(30,63),(30,64),(31,65),(31,68),(31,99),(32,68),(32,69),(32,93),(33,36),(33,42),(33,76),(34,118),(34,134),(35,121),(35,142),(36,118),(36,141),(37,3),(37,117),(38,116),(38,126),(39,115),(39,126),(40,121),(40,129),(41,114),(41,125),(42,118),(42,127),(43,117),(44,133),(45,132),(46,123),(47,128),(48,124),(49,125),(50,114),(51,6),(51,139),(52,2),(52,115),(53,16),(53,138),(54,119),(55,136),(56,84),(56,140),(57,60),(58,73),(58,135),(59,47),(59,140),(60,50),(61,51),(61,116),(62,56),(62,113),(63,9),(63,101),(64,71),(65,77),(65,137),(66,102),(67,38),(67,138),(68,89),(68,137),(69,90),(69,137),(70,57),(71,4),(71,65),(72,85),(72,129),(73,81),(73,123),(74,34),(74,122),(75,41),(75,130),(76,58),(76,127),(76,141),(77,25),(77,66),(78,60),(79,82),(79,120),(80,45),(80,115),(81,44),(81,114),(82,97),(83,37),(83,139),(84,91),(85,42),(85,131),(86,26),(86,72),(86,120),(87,24),(87,61),(87,138),(88,34),(88,36),(88,113),(89,79),(89,92),(90,39),(90,52),(91,41),(91,49),(92,35),(92,40),(92,120),(93,53),(93,67),(94,46),(94,73),(95,56),(95,59),(96,58),(96,94),(96,124),(97,48),(97,96),(98,57),(98,78),(99,21),(99,89),(99,112),(100,19),(100,90),(100,106),(101,69),(101,100),(102,62),(102,95),(102,142),(103,74),(103,88),(103,142),(104,37),(104,43),(105,35),(105,103),(105,136),(106,39),(106,80),(106,119),(107,40),(107,72),(107,136),(108,38),(108,61),(108,119),(109,76),(109,96),(109,131),(110,15),(110,93),(110,111),(111,67),(111,87),(111,108),(112,86),(112,92),(112,105),(112,107),(113,134),(113,140),(113,141),(114,133),(115,11),(115,132),(116,43),(116,139),(117,1),(118,143),(119,45),(119,126),(120,97),(120,109),(120,129),(121,122),(122,47),(122,134),(123,44),(124,46),(124,135),(125,133),(126,132),(127,135),(127,143),(128,49),(128,130),(129,48),(129,131),(130,125),(131,124),(131,127),(132,98),(134,128),(134,143),(135,123),(136,74),(136,121),(137,66),(138,83),(138,104),(138,116),(139,70),(139,117),(140,75),(140,91),(140,128),(141,75),(141,143),(142,59),(142,113),(142,122),(143,130)],144)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2 - 1
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St000640
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000640: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000640: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ? = 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ? = 2
[1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,2,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,3,4],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
=> ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,6),(0,7),(0,8),(1,4),(1,16),(2,5),(2,15),(3,2),(3,14),(4,10),(4,11),(5,9),(5,11),(6,12),(6,13),(7,12),(7,17),(8,3),(8,13),(8,17),(9,18),(10,18),(11,18),(12,19),(13,1),(13,19),(14,15),(15,9),(16,10),(17,14),(17,19),(19,16)],20)
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,2),(0,3),(0,10),(1,9),(1,11),(1,31),(2,1),(2,24),(3,6),(3,8),(3,30),(4,18),(4,27),(5,19),(5,20),(6,17),(6,29),(7,15),(7,21),(8,12),(8,26),(8,29),(9,16),(9,28),(10,24),(10,30),(11,14),(11,23),(11,28),(12,13),(12,22),(12,23),(13,32),(13,33),(14,33),(14,39),(15,37),(16,39),(17,38),(18,36),(19,34),(20,34),(21,5),(21,37),(22,21),(22,32),(23,25),(23,33),(24,31),(25,27),(25,35),(26,13),(26,38),(27,19),(27,36),(28,4),(28,25),(28,39),(29,7),(29,22),(29,38),(30,17),(30,26),(31,14),(31,16),(32,37),(33,35),(35,36),(36,34),(37,20),(38,15),(38,32),(39,18),(39,35)],40)
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,16),(0,17),(0,28),(1,23),(1,58),(2,19),(2,53),(3,11),(3,21),(3,80),(4,10),(4,20),(4,79),(5,12),(5,13),(5,81),(6,48),(6,73),(7,49),(7,74),(8,24),(8,75),(8,86),(9,25),(9,76),(9,87),(10,14),(10,71),(10,77),(11,15),(11,72),(11,78),(12,26),(12,69),(12,88),(13,27),(13,70),(13,88),(14,31),(14,82),(15,32),(15,83),(16,4),(16,36),(16,54),(17,3),(17,36),(17,55),(18,44),(18,67),(18,68),(19,39),(19,40),(19,43),(20,34),(20,56),(20,77),(21,35),(21,57),(21,78),(22,45),(22,46),(22,47),(23,18),(23,50),(23,82),(23,83),(24,41),(24,61),(24,63),(25,42),(25,62),(25,64),(26,37),(26,56),(26,84),(27,38),(27,57),(27,85),(28,54),(28,55),(28,81),(29,90),(29,118),(30,89),(30,117),(31,93),(32,94),(33,116),(33,124),(34,91),(34,122),(35,92),(35,123),(36,1),(36,121),(37,91),(37,110),(38,92),(38,111),(39,90),(39,104),(40,90),(40,105),(41,98),(41,108),(42,99),(42,109),(43,104),(43,105),(44,106),(44,107),(45,102),(45,103),(46,6),(46,89),(46,102),(47,7),(47,89),(47,103),(48,100),(49,101),(50,44),(50,93),(50,94),(51,39),(51,119),(52,40),(52,120),(53,43),(54,79),(54,121),(55,80),(55,121),(56,59),(56,91),(57,60),(57,92),(58,50),(59,86),(59,112),(60,87),(60,113),(61,65),(61,108),(62,66),(62,109),(63,51),(63,98),(64,52),(64,99),(65,29),(65,96),(66,29),(66,97),(67,61),(67,106),(67,124),(68,62),(68,107),(68,124),(69,37),(69,95),(70,38),(70,95),(71,31),(71,122),(72,32),(72,123),(73,51),(73,100),(74,52),(74,101),(75,41),(75,114),(76,42),(76,115),(77,8),(77,59),(77,122),(78,9),(78,60),(78,123),(79,34),(79,71),(80,35),(80,72),(81,69),(81,70),(82,33),(82,67),(82,93),(83,33),(83,68),(83,94),(84,30),(84,46),(84,110),(85,30),(85,47),(85,111),(86,63),(86,73),(86,114),(87,64),(87,74),(87,115),(88,22),(88,84),(88,85),(88,95),(89,2),(89,127),(90,125),(91,112),(92,113),(93,106),(93,116),(94,107),(94,116),(95,45),(95,110),(95,111),(96,118),(97,118),(98,119),(99,120),(100,119),(101,120),(102,48),(102,127),(103,49),(103,127),(104,125),(105,125),(106,108),(106,126),(107,109),(107,126),(108,96),(109,97),(110,102),(110,117),(111,103),(111,117),(112,114),(113,115),(114,98),(114,100),(115,99),(115,101),(116,126),(117,127),(118,125),(119,104),(120,105),(121,58),(122,75),(122,112),(123,76),(123,113),(124,65),(124,66),(124,126),(126,96),(126,97),(127,53)],128)
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,4),(0,8),(1,7),(1,21),(2,6),(2,20),(3,2),(3,18),(4,1),(4,19),(5,12),(5,13),(6,11),(6,16),(7,11),(7,17),(8,5),(8,18),(8,19),(9,23),(10,23),(11,22),(12,14),(13,15),(14,9),(15,10),(16,9),(16,22),(17,10),(17,22),(18,12),(18,20),(19,13),(19,21),(20,14),(20,16),(21,15),(21,17),(22,23)],24)
=> ? = 3
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,2),(0,3),(1,8),(1,12),(2,1),(3,4),(3,7),(4,25),(5,17),(6,24),(7,13),(7,25),(8,10),(8,28),(9,27),(10,9),(10,29),(11,22),(11,23),(12,21),(12,28),(13,18),(13,21),(14,30),(15,30),(17,6),(18,17),(19,16),(20,15),(20,31),(21,26),(22,24),(22,31),(23,14),(23,31),(24,19),(25,5),(25,18),(26,20),(26,22),(27,14),(27,15),(28,11),(28,26),(28,29),(29,20),(29,23),(29,27),(30,16),(31,19),(31,30)],32)
=> ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,9),(0,10),(1,2),(2,7),(2,11),(3,23),(4,17),(5,22),(6,13),(6,24),(7,8),(7,27),(8,26),(9,1),(9,21),(10,3),(10,21),(11,19),(11,27),(12,15),(12,20),(13,18),(13,19),(14,28),(15,28),(17,5),(18,17),(19,25),(20,22),(20,28),(21,6),(21,23),(22,16),(23,24),(24,4),(24,18),(25,14),(25,20),(26,14),(26,15),(27,12),(27,25),(27,26),(28,16)],29)
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,4),(0,14),(0,15),(1,26),(2,6),(2,27),(3,13),(3,28),(4,11),(4,12),(5,112),(6,5),(6,96),(7,32),(7,111),(8,23),(8,122),(9,24),(9,109),(10,25),(10,108),(11,22),(11,123),(12,21),(12,34),(12,123),(13,17),(13,20),(13,119),(14,2),(14,48),(15,3),(15,48),(16,93),(16,95),(17,120),(17,126),(18,90),(18,113),(19,35),(19,115),(19,118),(20,29),(20,120),(20,124),(21,94),(21,117),(22,97),(22,125),(23,89),(23,116),(24,87),(24,91),(25,88),(25,92),(26,30),(26,112),(26,126),(27,96),(27,114),(28,97),(28,119),(29,66),(29,106),(30,63),(30,99),(31,45),(31,79),(31,82),(32,44),(32,80),(32,81),(33,47),(33,57),(33,86),(34,94),(34,105),(34,114),(35,65),(35,71),(35,103),(36,131),(37,133),(38,130),(38,132),(39,132),(39,166),(40,133),(40,135),(41,127),(42,136),(42,143),(42,144),(43,129),(43,137),(44,129),(44,138),(45,134),(45,141),(46,10),(47,9),(47,166),(48,1),(49,153),(50,134),(50,135),(50,149),(51,139),(52,140),(52,159),(53,155),(54,143),(54,148),(55,144),(55,147),(56,150),(57,18),(57,130),(57,166),(58,44),(58,127),(59,76),(59,165),(60,80),(60,164),(61,70),(62,88),(62,139),(63,89),(63,165),(64,62),(65,60),(65,128),(66,107),(67,49),(68,53),(68,137),(69,51),(70,75),(71,73),(71,163),(72,108),(72,159),(73,43),(73,161),(74,42),(74,142),(75,41),(75,160),(76,37),(76,157),(77,36),(77,162),(78,40),(78,157),(78,163),(79,72),(79,141),(79,167),(80,110),(80,129),(81,62),(81,138),(82,83),(82,134),(82,167),(83,84),(83,151),(84,55),(84,152),(85,95),(85,146),(86,16),(86,85),(86,130),(87,69),(88,56),(88,136),(89,104),(90,46),(91,64),(92,49),(92,136),(93,58),(93,160),(94,61),(94,158),(95,87),(95,160),(96,8),(96,101),(97,102),(98,45),(98,50),(98,131),(99,71),(99,78),(99,165),(100,52),(100,72),(100,164),(101,70),(101,122),(102,77),(102,115),(103,50),(103,82),(103,128),(103,163),(104,43),(104,68),(105,38),(105,86),(105,158),(106,36),(106,98),(107,37),(107,40),(108,67),(108,92),(109,46),(109,91),(110,42),(110,54),(110,154),(111,64),(111,81),(112,59),(112,63),(113,60),(113,100),(114,61),(114,101),(115,65),(115,113),(115,162),(116,41),(116,58),(117,39),(117,47),(117,158),(118,31),(118,98),(118,103),(118,162),(119,19),(119,102),(119,124),(120,66),(120,121),(121,76),(121,78),(121,107),(122,75),(122,93),(122,116),(123,33),(123,105),(123,117),(123,125),(124,77),(124,106),(124,118),(125,38),(125,39),(125,57),(126,59),(126,99),(126,121),(127,51),(127,138),(128,149),(128,164),(128,167),(129,154),(130,7),(130,146),(131,52),(131,141),(131,149),(132,146),(133,53),(133,156),(134,151),(134,168),(135,151),(135,156),(136,150),(136,153),(137,55),(137,154),(137,155),(138,139),(139,56),(140,54),(140,142),(141,159),(141,168),(142,143),(143,153),(143,169),(144,150),(144,169),(146,111),(147,169),(148,169),(149,140),(149,168),(150,145),(151,152),(152,147),(153,145),(154,144),(154,148),(155,147),(155,148),(156,152),(156,155),(157,68),(157,133),(157,161),(158,85),(158,132),(159,67),(160,69),(160,127),(161,84),(161,137),(161,156),(162,79),(162,100),(162,128),(162,131),(163,83),(163,135),(163,161),(164,74),(164,110),(164,140),(165,73),(165,104),(165,157),(166,90),(166,109),(167,74),(167,168),(168,142),(169,145)],170)
=> ? = 3
Description
The rank of the largest boolean interval in a poset.
Matching statistic: St001632
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001632: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001632: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ? = 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ? = 2
[1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,2,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,3,4],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,4),(0,14),(0,15),(1,29),(2,6),(2,27),(3,13),(3,28),(4,10),(4,11),(5,102),(6,5),(6,87),(7,31),(7,99),(8,25),(8,98),(9,26),(9,72),(10,23),(10,109),(11,24),(11,33),(11,109),(12,22),(12,108),(13,17),(13,18),(13,107),(14,2),(14,38),(15,3),(15,38),(16,84),(16,85),(17,90),(17,111),(18,90),(18,101),(19,82),(19,83),(20,81),(20,100),(21,34),(21,89),(21,104),(22,86),(22,106),(23,91),(23,110),(24,80),(24,105),(25,78),(25,79),(26,52),(26,88),(27,87),(27,103),(28,91),(28,107),(29,30),(29,102),(29,111),(30,63),(30,96),(31,44),(31,71),(31,76),(32,37),(32,51),(32,77),(33,80),(33,94),(33,103),(34,62),(34,74),(34,75),(35,133),(36,9),(37,8),(37,127),(38,1),(39,118),(40,117),(40,122),(41,119),(42,119),(42,126),(43,116),(43,127),(44,112),(44,120),(45,112),(45,113),(46,114),(46,116),(47,121),(48,122),(49,125),(50,113),(51,20),(51,114),(51,127),(52,117),(53,72),(54,88),(54,121),(55,45),(55,132),(56,48),(57,84),(57,124),(58,76),(58,134),(59,66),(60,56),(61,54),(62,55),(62,126),(63,86),(63,129),(64,53),(64,134),(65,47),(66,70),(67,40),(68,40),(68,123),(69,41),(69,129),(70,39),(70,128),(71,54),(71,120),(72,52),(73,44),(73,118),(74,58),(74,115),(75,82),(75,115),(75,126),(76,68),(76,112),(77,16),(77,57),(77,114),(78,65),(79,61),(80,59),(80,130),(81,36),(82,60),(82,131),(83,53),(83,131),(84,78),(84,128),(85,73),(85,128),(86,92),(87,12),(87,95),(88,49),(88,117),(89,19),(89,75),(89,133),(90,97),(91,93),(92,45),(92,50),(93,35),(93,104),(94,46),(94,77),(94,130),(95,66),(95,108),(96,42),(96,62),(96,129),(97,41),(97,42),(98,36),(98,79),(99,61),(99,71),(100,58),(100,64),(101,35),(101,89),(102,63),(102,69),(103,59),(103,95),(104,74),(104,100),(104,133),(105,37),(105,43),(105,130),(106,39),(106,73),(107,21),(107,93),(107,101),(108,70),(108,85),(108,106),(109,32),(109,94),(109,105),(109,110),(110,43),(110,46),(110,51),(111,69),(111,96),(111,97),(112,123),(113,48),(113,123),(114,7),(114,124),(115,131),(115,134),(116,124),(117,125),(118,47),(118,120),(119,50),(119,132),(120,121),(121,49),(122,125),(123,122),(124,99),(126,60),(126,132),(127,81),(127,98),(128,65),(128,118),(129,55),(129,92),(129,119),(130,57),(130,116),(131,67),(132,56),(132,113),(133,64),(133,83),(133,115),(134,67),(134,68)],135)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,15),(0,16),(0,30),(1,26),(2,14),(2,95),(3,78),(4,79),(5,92),(6,77),(7,29),(7,93),(8,27),(8,96),(9,28),(9,94),(10,12),(10,32),(10,80),(11,84),(12,13),(12,86),(13,11),(13,97),(14,19),(14,98),(15,10),(15,54),(16,2),(16,24),(16,76),(17,47),(17,71),(18,46),(18,72),(19,41),(19,91),(20,35),(20,73),(21,69),(21,87),(22,25),(22,75),(22,90),(23,45),(23,70),(24,31),(24,88),(24,95),(25,49),(25,50),(26,39),(26,40),(27,52),(27,53),(28,41),(28,89),(29,33),(29,57),(30,54),(30,76),(31,67),(31,68),(31,85),(32,56),(32,68),(32,86),(33,110),(34,106),(34,107),(35,21),(35,111),(36,100),(37,101),(37,106),(38,102),(39,102),(40,102),(41,99),(42,104),(43,113),(44,103),(45,6),(45,112),(46,7),(46,108),(47,5),(47,107),(48,3),(48,103),(49,55),(49,105),(50,33),(50,105),(51,61),(52,63),(53,61),(54,80),(55,52),(55,113),(56,79),(56,109),(57,60),(57,110),(58,90),(59,81),(60,62),(61,39),(62,40),(63,38),(64,38),(65,37),(65,111),(66,36),(66,108),(67,82),(67,109),(68,83),(68,109),(69,59),(70,48),(70,112),(71,42),(71,107),(72,69),(72,108),(73,23),(73,74),(73,111),(74,45),(74,101),(75,50),(75,100),(76,9),(76,88),(77,51),(78,62),(79,22),(79,58),(80,4),(80,56),(81,60),(81,78),(82,34),(82,47),(83,66),(83,72),(84,36),(84,75),(85,35),(85,65),(86,18),(86,83),(86,97),(87,44),(87,48),(88,67),(88,94),(89,34),(89,71),(89,99),(90,49),(90,92),(90,100),(91,37),(91,74),(91,99),(92,43),(92,55),(93,57),(93,81),(94,17),(94,82),(94,89),(95,20),(95,85),(95,98),(96,51),(96,53),(97,46),(97,66),(97,84),(98,65),(98,73),(98,91),(99,42),(99,106),(100,43),(100,105),(101,44),(101,112),(103,1),(104,96),(105,110),(105,113),(106,104),(107,8),(107,104),(108,59),(108,93),(109,58),(110,64),(111,70),(111,87),(111,101),(112,77),(112,103),(113,63),(113,64)],114)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
=> ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,6),(0,7),(0,8),(1,4),(1,16),(2,5),(2,15),(3,2),(3,14),(4,10),(4,11),(5,9),(5,11),(6,12),(6,13),(7,12),(7,17),(8,3),(8,13),(8,17),(9,18),(10,18),(11,18),(12,19),(13,1),(13,19),(14,15),(15,9),(16,10),(17,14),(17,19),(19,16)],20)
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,2),(0,3),(0,10),(1,9),(1,11),(1,31),(2,1),(2,24),(3,6),(3,8),(3,30),(4,18),(4,27),(5,19),(5,20),(6,17),(6,29),(7,15),(7,21),(8,12),(8,26),(8,29),(9,16),(9,28),(10,24),(10,30),(11,14),(11,23),(11,28),(12,13),(12,22),(12,23),(13,32),(13,33),(14,33),(14,39),(15,37),(16,39),(17,38),(18,36),(19,34),(20,34),(21,5),(21,37),(22,21),(22,32),(23,25),(23,33),(24,31),(25,27),(25,35),(26,13),(26,38),(27,19),(27,36),(28,4),(28,25),(28,39),(29,7),(29,22),(29,38),(30,17),(30,26),(31,14),(31,16),(32,37),(33,35),(35,36),(36,34),(37,20),(38,15),(38,32),(39,18),(39,35)],40)
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,16),(0,17),(0,28),(1,23),(1,58),(2,19),(2,53),(3,11),(3,21),(3,80),(4,10),(4,20),(4,79),(5,12),(5,13),(5,81),(6,48),(6,73),(7,49),(7,74),(8,24),(8,75),(8,86),(9,25),(9,76),(9,87),(10,14),(10,71),(10,77),(11,15),(11,72),(11,78),(12,26),(12,69),(12,88),(13,27),(13,70),(13,88),(14,31),(14,82),(15,32),(15,83),(16,4),(16,36),(16,54),(17,3),(17,36),(17,55),(18,44),(18,67),(18,68),(19,39),(19,40),(19,43),(20,34),(20,56),(20,77),(21,35),(21,57),(21,78),(22,45),(22,46),(22,47),(23,18),(23,50),(23,82),(23,83),(24,41),(24,61),(24,63),(25,42),(25,62),(25,64),(26,37),(26,56),(26,84),(27,38),(27,57),(27,85),(28,54),(28,55),(28,81),(29,90),(29,118),(30,89),(30,117),(31,93),(32,94),(33,116),(33,124),(34,91),(34,122),(35,92),(35,123),(36,1),(36,121),(37,91),(37,110),(38,92),(38,111),(39,90),(39,104),(40,90),(40,105),(41,98),(41,108),(42,99),(42,109),(43,104),(43,105),(44,106),(44,107),(45,102),(45,103),(46,6),(46,89),(46,102),(47,7),(47,89),(47,103),(48,100),(49,101),(50,44),(50,93),(50,94),(51,39),(51,119),(52,40),(52,120),(53,43),(54,79),(54,121),(55,80),(55,121),(56,59),(56,91),(57,60),(57,92),(58,50),(59,86),(59,112),(60,87),(60,113),(61,65),(61,108),(62,66),(62,109),(63,51),(63,98),(64,52),(64,99),(65,29),(65,96),(66,29),(66,97),(67,61),(67,106),(67,124),(68,62),(68,107),(68,124),(69,37),(69,95),(70,38),(70,95),(71,31),(71,122),(72,32),(72,123),(73,51),(73,100),(74,52),(74,101),(75,41),(75,114),(76,42),(76,115),(77,8),(77,59),(77,122),(78,9),(78,60),(78,123),(79,34),(79,71),(80,35),(80,72),(81,69),(81,70),(82,33),(82,67),(82,93),(83,33),(83,68),(83,94),(84,30),(84,46),(84,110),(85,30),(85,47),(85,111),(86,63),(86,73),(86,114),(87,64),(87,74),(87,115),(88,22),(88,84),(88,85),(88,95),(89,2),(89,127),(90,125),(91,112),(92,113),(93,106),(93,116),(94,107),(94,116),(95,45),(95,110),(95,111),(96,118),(97,118),(98,119),(99,120),(100,119),(101,120),(102,48),(102,127),(103,49),(103,127),(104,125),(105,125),(106,108),(106,126),(107,109),(107,126),(108,96),(109,97),(110,102),(110,117),(111,103),(111,117),(112,114),(113,115),(114,98),(114,100),(115,99),(115,101),(116,126),(117,127),(118,125),(119,104),(120,105),(121,58),(122,75),(122,112),(123,76),(123,113),(124,65),(124,66),(124,126),(126,96),(126,97),(127,53)],128)
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,3),(0,4),(0,8),(1,7),(1,21),(2,6),(2,20),(3,2),(3,18),(4,1),(4,19),(5,12),(5,13),(6,11),(6,16),(7,11),(7,17),(8,5),(8,18),(8,19),(9,23),(10,23),(11,22),(12,14),(13,15),(14,9),(15,10),(16,9),(16,22),(17,10),(17,22),(18,12),(18,20),(19,13),(19,21),(20,14),(20,16),(21,15),(21,17),(22,23)],24)
=> ? = 3
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]]
=> ([(0,2),(0,3),(1,8),(1,12),(2,1),(3,4),(3,7),(4,25),(5,17),(6,24),(7,13),(7,25),(8,10),(8,28),(9,27),(10,9),(10,29),(11,22),(11,23),(12,21),(12,28),(13,18),(13,21),(14,30),(15,30),(17,6),(18,17),(19,16),(20,15),(20,31),(21,26),(22,24),(22,31),(23,14),(23,31),(24,19),(25,5),(25,18),(26,20),(26,22),(27,14),(27,15),(28,11),(28,26),(28,29),(29,20),(29,23),(29,27),(30,16),(31,19),(31,30)],32)
=> ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,3),(0,4),(0,7),(1,5),(1,17),(2,6),(2,8),(2,18),(3,1),(3,13),(4,2),(4,14),(5,10),(5,15),(6,11),(6,16),(7,13),(7,14),(8,12),(8,15),(8,16),(9,21),(10,19),(11,20),(12,19),(12,20),(13,17),(14,18),(15,9),(15,19),(16,9),(16,20),(17,10),(18,11),(18,12),(19,21),(20,21)],22)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]]
=> ([(0,9),(0,10),(1,2),(2,7),(2,11),(3,23),(4,17),(5,22),(6,13),(6,24),(7,8),(7,27),(8,26),(9,1),(9,21),(10,3),(10,21),(11,19),(11,27),(12,15),(12,20),(13,18),(13,19),(14,28),(15,28),(17,5),(18,17),(19,25),(20,22),(20,28),(21,6),(21,23),(22,16),(23,24),(24,4),(24,18),(25,14),(25,20),(26,14),(26,15),(27,12),(27,25),(27,26),(28,16)],29)
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]]
=> ([(0,4),(0,14),(0,15),(1,26),(2,6),(2,27),(3,13),(3,28),(4,11),(4,12),(5,112),(6,5),(6,96),(7,32),(7,111),(8,23),(8,122),(9,24),(9,109),(10,25),(10,108),(11,22),(11,123),(12,21),(12,34),(12,123),(13,17),(13,20),(13,119),(14,2),(14,48),(15,3),(15,48),(16,93),(16,95),(17,120),(17,126),(18,90),(18,113),(19,35),(19,115),(19,118),(20,29),(20,120),(20,124),(21,94),(21,117),(22,97),(22,125),(23,89),(23,116),(24,87),(24,91),(25,88),(25,92),(26,30),(26,112),(26,126),(27,96),(27,114),(28,97),(28,119),(29,66),(29,106),(30,63),(30,99),(31,45),(31,79),(31,82),(32,44),(32,80),(32,81),(33,47),(33,57),(33,86),(34,94),(34,105),(34,114),(35,65),(35,71),(35,103),(36,131),(37,133),(38,130),(38,132),(39,132),(39,166),(40,133),(40,135),(41,127),(42,136),(42,143),(42,144),(43,129),(43,137),(44,129),(44,138),(45,134),(45,141),(46,10),(47,9),(47,166),(48,1),(49,153),(50,134),(50,135),(50,149),(51,139),(52,140),(52,159),(53,155),(54,143),(54,148),(55,144),(55,147),(56,150),(57,18),(57,130),(57,166),(58,44),(58,127),(59,76),(59,165),(60,80),(60,164),(61,70),(62,88),(62,139),(63,89),(63,165),(64,62),(65,60),(65,128),(66,107),(67,49),(68,53),(68,137),(69,51),(70,75),(71,73),(71,163),(72,108),(72,159),(73,43),(73,161),(74,42),(74,142),(75,41),(75,160),(76,37),(76,157),(77,36),(77,162),(78,40),(78,157),(78,163),(79,72),(79,141),(79,167),(80,110),(80,129),(81,62),(81,138),(82,83),(82,134),(82,167),(83,84),(83,151),(84,55),(84,152),(85,95),(85,146),(86,16),(86,85),(86,130),(87,69),(88,56),(88,136),(89,104),(90,46),(91,64),(92,49),(92,136),(93,58),(93,160),(94,61),(94,158),(95,87),(95,160),(96,8),(96,101),(97,102),(98,45),(98,50),(98,131),(99,71),(99,78),(99,165),(100,52),(100,72),(100,164),(101,70),(101,122),(102,77),(102,115),(103,50),(103,82),(103,128),(103,163),(104,43),(104,68),(105,38),(105,86),(105,158),(106,36),(106,98),(107,37),(107,40),(108,67),(108,92),(109,46),(109,91),(110,42),(110,54),(110,154),(111,64),(111,81),(112,59),(112,63),(113,60),(113,100),(114,61),(114,101),(115,65),(115,113),(115,162),(116,41),(116,58),(117,39),(117,47),(117,158),(118,31),(118,98),(118,103),(118,162),(119,19),(119,102),(119,124),(120,66),(120,121),(121,76),(121,78),(121,107),(122,75),(122,93),(122,116),(123,33),(123,105),(123,117),(123,125),(124,77),(124,106),(124,118),(125,38),(125,39),(125,57),(126,59),(126,99),(126,121),(127,51),(127,138),(128,149),(128,164),(128,167),(129,154),(130,7),(130,146),(131,52),(131,141),(131,149),(132,146),(133,53),(133,156),(134,151),(134,168),(135,151),(135,156),(136,150),(136,153),(137,55),(137,154),(137,155),(138,139),(139,56),(140,54),(140,142),(141,159),(141,168),(142,143),(143,153),(143,169),(144,150),(144,169),(146,111),(147,169),(148,169),(149,140),(149,168),(150,145),(151,152),(152,147),(153,145),(154,144),(154,148),(155,147),(155,148),(156,152),(156,155),(157,68),(157,133),(157,161),(158,85),(158,132),(159,67),(160,69),(160,127),(161,84),(161,137),(161,156),(162,79),(162,100),(162,128),(162,131),(163,83),(163,135),(163,161),(164,74),(164,110),(164,140),(165,73),(165,104),(165,157),(166,90),(166,109),(167,74),(167,168),(168,142),(169,145)],170)
=> ? = 3
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001729
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00329: Permutations —Tanimoto⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,2,1,3,5] => 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,1,2,4,5] => 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,5,2,1,4] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,2,3,1,4,6] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [2,3,6,4,5,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [5,2,3,6,1,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,2,1,3,5,6] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,2,6,3,1,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [2,5,3,4,6,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,2,5,1,3,6] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [2,5,6,3,4,1] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [3,1,2,4,5,6] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,6,2,4,1,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [3,5,2,1,4,6] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [3,1,2,6,4,5] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,5,2,6,1,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [4,2,3,5,6,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,6,3,4,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [3,4,1,2,5,6] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [3,4,6,2,1,5] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [4,5,2,3,6,1] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [4,6,2,3,5,1] => 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6,4,5,2,1,3] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [3,1,5,6,2,4] => 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [7,2,3,4,5,1,6] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [6,2,3,4,1,5,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [2,3,4,7,5,6,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [6,2,3,4,7,1,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [5,2,3,1,4,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [5,2,3,7,4,1,6] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [2,3,6,4,5,7,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [2,3,7,4,5,6,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [7,2,3,6,4,1,5] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [5,2,3,6,1,4,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [5,2,3,1,7,4,6] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [2,3,6,7,4,5,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [5,2,3,6,7,1,4] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [4,2,1,3,5,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [4,2,7,3,5,1,6] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [4,2,6,3,1,5,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [4,2,1,3,7,5,6] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [4,2,6,3,7,1,5] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [2,5,3,4,6,7,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [7,2,5,3,4,1,6] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [2,6,3,4,5,7,1] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [6,2,5,3,1,4,7] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [2,6,3,7,4,5,1] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [4,2,5,1,3,6,7] => ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [4,2,5,7,3,1,6] => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [4,2,1,6,3,5,7] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [2,5,6,3,4,7,1] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [2,5,7,3,4,6,1] => ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [7,2,5,6,3,1,4] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [4,2,5,6,1,3,7] => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [4,2,1,6,7,3,5] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [3,1,2,4,5,6,7] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [3,7,2,4,5,1,6] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [3,6,2,4,1,5,7] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [3,1,2,4,7,5,6] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [3,6,2,4,7,1,5] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [3,5,2,1,4,6,7] => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [3,5,2,7,4,1,6] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [3,1,2,6,4,5,7] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [3,1,2,7,4,5,6] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [3,7,2,6,4,1,5] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [3,5,2,6,1,4,7] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [3,5,2,1,7,4,6] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [3,1,2,6,7,4,5] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [3,5,2,6,7,1,4] => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [4,2,3,5,6,7,1] => ? = 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [7,4,2,3,5,1,6] => ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [6,4,2,3,1,5,7] => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [4,2,3,7,5,6,1] => ? = 3
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Matching statistic: St000353
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ? = 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 3
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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