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Your data matches 68 different statistics following compositions of up to 3 maps.
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Matching statistic: St000966
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000966: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> 0
Description
Number of peaks minus the global dimension of the corresponding LNakayama algebra.
Matching statistic: St000455
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [2,1] => [1,2] => ([],2)
=> ? = 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> 0
[1,1,0,0]
=> [2,3,1] => [1,2,3] => ([],3)
=> ? = 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [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
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,6,5,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,6,4,2,3,5] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,6,3,4,2,5] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [1,6,7,5,4,3,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [1,5,7,6,4,3,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [1,7,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [1,5,6,7,4,3,2] => ([(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
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [1,4,7,6,5,3,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [1,4,6,7,5,3,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [1,7,6,4,5,3,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,7,5,3,2,4,6] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [1,6,7,4,5,3,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [1,4,5,7,6,3,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [1,4,7,5,6,3,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [1,7,4,5,6,3,2] => ([(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
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [1,3,7,6,5,4,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [1,3,6,7,5,4,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [1,3,5,7,6,4,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [1,3,7,5,6,4,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [1,3,5,6,7,4,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [1,7,6,4,2,3,5] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [1,6,4,2,3,7,5] => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [1,6,7,4,2,3,5] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [1,5,7,6,3,4,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [1,7,4,2,3,5,6] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [1,3,4,7,6,5,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [1,3,4,6,7,5,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [1,3,7,6,4,5,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => ([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [1,3,6,7,4,5,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [1,7,5,2,3,4,6] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [1,3,4,5,7,6,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [1,3,4,7,5,6,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [1,3,7,4,5,6,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [1,2,5,7,6,4,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [1,2,4,7,6,5,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [1,2,4,6,7,5,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 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: St001875
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,0,1,0]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,0]
=> [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ?
=> ?
=> ? = 0 + 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,14),(3,14),(4,9),(5,8),(6,8),(6,10),(7,9),(7,10),(8,11),(9,12),(10,11),(10,12),(11,13),(12,13),(13,14)],15)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,11),(4,11),(5,9),(6,8),(7,8),(7,9),(8,10),(9,10),(10,11)],12)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,15),(3,15),(4,13),(5,8),(6,14),(6,16),(7,12),(7,16),(9,11),(10,11),(11,8),(12,10),(13,12),(14,9),(15,13),(16,9),(16,10)],17)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,12),(3,9),(4,8),(5,10),(6,10),(7,8),(7,9),(8,11),(9,11),(10,12),(11,12)],13)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8),(8,10),(9,10)],11)
=> ([(0,1)],2)
=> ? = 2 + 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,13),(2,16),(3,14),(4,14),(5,12),(6,8),(7,13),(7,15),(9,11),(10,11),(11,8),(12,10),(13,9),(14,16),(15,9),(15,10),(16,12),(16,15)],17)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,15),(2,14),(3,13),(4,12),(5,8),(6,14),(6,15),(7,11),(7,13),(9,12),(10,8),(11,10),(12,11),(13,10),(14,9),(15,9)],16)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8),(8,10),(9,10)],11)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8),(7,12),(8,14),(9,13),(10,13),(12,14),(13,12),(14,11)],15)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,12),(3,9),(4,8),(5,10),(6,10),(7,8),(7,9),(8,11),(9,11),(10,12),(11,12)],13)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,11),(4,11),(5,9),(6,8),(7,8),(7,9),(8,10),(9,10),(10,11)],12)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,13),(2,16),(3,14),(4,14),(5,12),(6,8),(7,13),(7,15),(9,11),(10,11),(11,8),(12,10),(13,9),(14,16),(15,9),(15,10),(16,12),(16,15)],17)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,11),(3,14),(4,13),(5,15),(6,8),(7,13),(7,14),(9,15),(10,8),(11,10),(12,10),(13,9),(14,9),(15,11),(15,12)],16)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,10),(3,10),(4,10),(5,10),(6,10),(7,8),(8,9),(10,8)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,9),(3,8),(4,8),(5,11),(6,10),(7,12),(8,14),(9,14),(10,13),(11,13),(13,12),(14,10),(14,11)],15)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,13),(3,15),(4,12),(5,8),(6,13),(6,14),(7,11),(7,15),(9,11),(10,12),(11,10),(12,8),(13,9),(14,9),(15,10)],16)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,11),(3,10),(4,9),(5,8),(6,8),(7,9),(7,10),(8,14),(9,13),(10,13),(11,12),(13,14),(14,11)],15)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,15),(3,15),(4,13),(5,8),(6,14),(6,16),(7,12),(7,16),(9,11),(10,11),(11,8),(12,10),(13,12),(14,9),(15,13),(16,9),(16,10)],17)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,9),(4,9),(5,12),(6,11),(7,10),(8,11),(9,12),(11,13),(12,13),(13,10)],14)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,15),(2,14),(3,13),(4,12),(5,8),(6,14),(6,15),(7,11),(7,13),(9,12),(10,8),(11,10),(12,11),(13,10),(14,9),(15,9)],16)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,10),(3,10),(4,10),(5,10),(6,10),(7,8),(8,9),(10,8)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,9),(4,9),(5,8),(6,8),(7,10),(8,11),(9,11),(11,10)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8),(7,12),(8,14),(9,13),(10,13),(12,14),(13,12),(14,11)],15)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001208
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> []
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> [1]
=> [1,0]
=> [2,1] => 1 = 0 + 1
[1,1,0,0]
=> []
=> []
=> [1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [2,1] => 1 = 0 + 1
[1,1,1,0,0,0]
=> []
=> []
=> [1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [10,1,4,6,2,7,9,3,5,8] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [9,3,5,1,6,8,2,4,7] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [4,1,2,10,6,7,9,3,5,8] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [3,1,9,5,6,8,2,4,7] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,8,4,5,7,1,3,6] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [7,1,10,6,2,3,9,4,5,8] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,9,5,1,2,8,3,4,7] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [10,1,2,7,6,3,9,4,5,8] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [9,1,6,5,2,8,3,4,7] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [8,5,4,1,7,2,3,6] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [10,1,2,3,6,7,9,4,5,8] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [9,1,2,5,6,8,3,4,7] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [8,1,4,5,7,2,3,6] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,1,4,5,2,10,9,3,7,8] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,3,4,1,9,8,2,6,7] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,10,9,3,7,8] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [3,1,4,5,9,8,2,6,7] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,4,8,7,1,5,6] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [10,1,6,5,2,3,9,4,7,8] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [9,5,4,1,2,8,3,6,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [10,1,2,5,6,3,9,4,7,8] => ? = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1 = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001236
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> => [1] => 1 = 0 + 1
[1,0,1,0]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> []
=> => [1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => [1,1,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2]
=> 100 => [1,3] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[1,1,1,0,0,0]
=> []
=> => [1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => [1,2,2,2] => ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => [1,1,1,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => [1,1,3] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => [1,1,2] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => [1,4] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => [1,3] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> []
=> => [1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => [1,2,2,2,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => [1,1,2,2,2] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => [1,3,1,2,2] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => [1,2,1,2,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => [1,1,1,2,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => [1,2,3,1,2] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => [1,1,3,1,2] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => [1,3,2,1,2] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => [1,2,2,1,2] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => [1,1,2,1,2] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => [1,4,1,1,2] => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => [1,1,1,1,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => [1,2,2,3] => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => [1,1,2,3] => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => [1,3,1,3] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => [1,2,1,3] => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => [1,1,1,3] => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => [1,2,3,2] => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => [1,1,3,2] => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => [1,3,2,2] => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => [1,2,2,2] => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => [1,1,1,2] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => [1,2,4] => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => [1,1,4] => 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => [1,3,3] => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => [1,1,3] => 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => [1,4,2] => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 110 => [1,1,2] => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 10000 => [1,5] => 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1000 => [1,4] => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 100 => [1,3] => 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => [1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 1010101010 => [1,2,2,2,2,2] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 110101010 => [1,1,2,2,2,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> 1001101010 => [1,3,1,2,2,2] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 101101010 => [1,2,1,2,2,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 11101010 => [1,1,1,2,2,2] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> 1010011010 => [1,2,3,1,2,2] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> 110011010 => [1,1,3,1,2,2] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> 1001011010 => [1,3,2,1,2,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 101011010 => [1,2,2,1,2,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 11011010 => [1,1,2,1,2,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 1000111010 => [1,4,1,1,2,2] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> 100111010 => [1,3,1,1,2,2] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> 10111010 => [1,2,1,1,2,2] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 1111010 => [1,1,1,1,2,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> 1010100110 => [1,2,2,3,1,2] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> 110100110 => [1,1,2,3,1,2] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 1001100110 => [1,3,1,3,1,2] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> 101100110 => [1,2,1,3,1,2] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> 11100110 => [1,1,1,3,1,2] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> 1010010110 => [1,2,3,2,1,2] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> 110010110 => [1,1,3,2,1,2] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> 1001010110 => [1,3,2,2,1,2] => ? = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 11110 => [1,1,1,1,2] => 1 = 0 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 11100 => [1,1,1,3] => 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 1110 => [1,1,1,2] => 1 = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 11000 => [1,1,4] => 1 = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1100 => [1,1,3] => 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 110 => [1,1,2] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 10000 => [1,5] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1000 => [1,4] => 1 = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 100 => [1,3] => 1 = 0 + 1
Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
Matching statistic: St001371
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> []
=> => ? = 0
[1,0,1,0]
=> [1]
=> [1,0]
=> 10 => 0
[1,1,0,0]
=> []
=> []
=> => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 0
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 10 => 0
[1,1,1,0,0,0]
=> []
=> []
=> => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 10 => 0
[1,1,1,1,0,0,0,0]
=> []
=> []
=> => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 10111011000100 => ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 111011000100 => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 10101111000100 => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => ? = 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? = 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> 10111010010100 => ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 111010010100 => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 10101110010100 => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 101110100100 => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? = 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 101011100100 => ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> 10 => 0
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> 101110111001000100 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> 1110111001000100 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 101011111001000100 => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 1011111001000100 => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 11111001000100 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> 101110101101000100 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> 1110101101000100 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> 101011101101000100 => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> 1011101101000100 => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> 11101101000100 => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> 101010111101000100 => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> 1010111101000100 => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> 10111101000100 => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 111101000100 => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> 101110111000010100 => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> 1110111000010100 => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 101011111000010100 => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 0
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1,0]
=> 10 => 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001730
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> []
=> => ? = 0
[1,0,1,0]
=> [1]
=> [1,0]
=> 10 => 0
[1,1,0,0]
=> []
=> []
=> => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 0
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 10 => 0
[1,1,1,0,0,0]
=> []
=> []
=> => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 10 => 0
[1,1,1,1,0,0,0,0]
=> []
=> []
=> => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 10111011000100 => ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 111011000100 => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 10101111000100 => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => ? = 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? = 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> 10111010010100 => ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 111010010100 => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 10101110010100 => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 101110110000 => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 101110100100 => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? = 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 101011100100 => ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> 10 => 0
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> 101110111001000100 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> 1110111001000100 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 101011111001000100 => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 1011111001000100 => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 11111001000100 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> 101110101101000100 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> 1110101101000100 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> 101011101101000100 => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> 1011101101000100 => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> 11101101000100 => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> 101010111101000100 => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> 1010111101000100 => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> 10111101000100 => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 111101000100 => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> 101110111000010100 => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> 1110111000010100 => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 101011111000010100 => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> 1010 => 0
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1,0]
=> 10 => 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001195
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> []
=> [1,0]
=> ? = 0 + 1
[1,0,1,0]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> ? = 0 + 1
[1,1,0,0]
=> []
=> []
=> [1,0]
=> ? = 0 + 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> ? = 0 + 1
[1,1,1,0,0,0]
=> []
=> []
=> [1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> ? = 0 + 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001845
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> ([],1)
=> ([(0,1)],2)
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ([(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)
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 0
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ?
=> ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
=> ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ? = 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
=> ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
=> ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([],6)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,1),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,6)],18)
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(2,7),(3,8),(3,9),(3,10),(4,10),(4,12),(4,13),(5,9),(5,11),(5,13),(6,8),(6,11),(6,12),(7,1),(8,14),(8,15),(9,14),(9,16),(10,15),(10,16),(11,14),(11,17),(12,15),(12,17),(13,16),(13,17),(14,18),(15,18),(16,18),(17,2),(17,18),(18,7)],19)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5)],6)
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,11),(3,12),(3,13),(4,9),(4,10),(4,13),(5,8),(5,10),(5,12),(6,8),(6,9),(6,11),(8,14),(8,17),(9,14),(9,15),(10,14),(10,16),(11,15),(11,17),(12,16),(12,17),(13,15),(13,16),(14,18),(15,18),(16,18),(17,18),(18,1),(18,2)],19)
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,14),(3,8),(3,9),(3,10),(4,10),(4,12),(4,13),(5,9),(5,11),(5,13),(6,8),(6,11),(6,12),(8,15),(8,16),(9,15),(9,17),(10,16),(10,17),(11,15),(11,18),(12,16),(12,18),(13,17),(13,18),(14,7),(15,19),(16,19),(17,19),(18,2),(18,19),(19,1),(19,14)],20)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(1,3),(1,4),(1,5),(1,6),(2,7),(3,8),(3,9),(3,10),(4,10),(4,12),(4,13),(5,9),(5,11),(5,13),(6,8),(6,11),(6,12),(8,14),(8,15),(9,14),(9,16),(10,15),(10,16),(11,14),(11,17),(12,15),(12,17),(13,16),(13,17),(14,18),(15,18),(16,18),(17,2),(17,18),(18,7)],19)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> ([(0,5),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,6),(6,1),(6,2),(6,3),(6,4),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17)],18)
=> ? = 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]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
=> 0
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=> 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> 0
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> 0
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 0
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> 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]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,7),(2,8),(3,8),(4,5),(5,1),(6,4),(7,2),(7,3),(8,6)],9)
=> 0
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
Matching statistic: St001550
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,2,6,4,3,5] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,2,5,6,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,6,4,2,3,5] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,6,3,4,2,5] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,5,6,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,2,3,6,4,5] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,2,6,3,4,5] => 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,6,2,3,4,5] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [1,7,5,3,2,6,4] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [1,5,3,2,6,7,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [1,7,5,6,3,2,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [1,5,6,3,2,7,4] => ? = 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [1,5,6,7,3,2,4] => ? = 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [1,7,5,2,3,4,6] => ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => [1,7,5,2,4,6,3] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [1,6,3,2,7,4,5] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [1,6,7,3,2,4,5] => ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => [1,4,7,5,6,2,3] => ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [1,7,3,2,4,5,6] => ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [1,2,6,4,3,7,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [1,2,6,7,4,3,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => [1,2,7,5,6,3,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [1,2,7,4,3,5,6] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,2,5,6,7,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [1,6,4,2,3,7,5] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [1,6,7,4,2,3,5] => ? = 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [1,6,3,4,2,7,5] => ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => [1,6,3,5,2,7,4] => ? = 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [1,6,7,3,4,2,5] => ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => [1,7,5,6,2,3,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [1,7,4,2,3,5,6] => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [1,7,3,4,2,5,6] => ? = 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => [1,5,6,7,2,3,4] => ? = 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [1,2,3,7,5,4,6] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [1,2,3,6,7,4,5] => 0
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [1,2,7,5,3,4,6] => 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [1,2,7,4,5,3,6] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [1,2,6,7,3,4,5] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [1,7,5,2,3,4,6] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [1,7,4,5,2,3,6] => ? = 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [1,7,3,4,5,2,6] => ? = 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => [1,6,7,2,3,4,5] => ? = 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [1,2,3,4,7,5,6] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => [1,2,3,7,4,5,6] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => [1,2,7,3,4,5,6] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => [1,7,2,3,4,5,6] => ? = 0
[1,1,1,1,1,0,0,0,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => [1,8,6,4,2,7,5,3] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,6,1,2,3,4,8,5] => [1,7,8,5,3,2,6,4] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => [1,5,3,2,8,6,7,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,6,1,2,3,7,4,5] => [1,8,5,3,2,6,7,4] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => [1,5,3,2,6,7,8,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [8,4,1,2,7,3,5,6] => [1,8,6,3,2,4,5,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [7,4,1,2,6,3,8,5] => [1,7,8,5,6,3,2,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [5,8,1,2,7,3,4,6] => [1,5,7,4,2,8,6,3] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [8,7,1,2,6,3,4,5] => [1,8,5,6,3,2,7,4] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,7,1,2,6,3,8,4] => [1,5,6,3,2,7,8,4] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [5,4,1,2,8,7,3,6] => [1,5,8,6,7,3,2,4] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [8,4,1,2,6,7,3,5] => [1,8,5,6,7,3,2,4] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,8,1,2,6,7,3,4] => [1,5,6,7,3,2,8,4] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,4,1,2,6,7,8,3] => [1,5,6,7,8,3,2,4] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [7,3,1,8,2,4,5,6] => [1,7,5,2,3,4,8,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [7,3,1,6,2,4,8,5] => [1,7,8,5,2,3,4,6] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [8,3,1,5,2,7,4,6] => [1,8,6,7,4,5,2,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [8,3,1,6,2,7,4,5] => [1,8,5,2,3,4,6,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [6,3,1,5,2,7,8,4] => [1,6,7,8,4,5,2,3] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [7,4,1,8,2,3,5,6] => [1,7,5,2,4,8,6,3] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [7,4,1,6,2,3,8,5] => [1,7,8,5,2,4,6,3] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [8,7,1,5,2,3,4,6] => [1,8,6,3,2,7,4,5] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [8,7,1,6,2,3,4,5] => [1,8,5,2,7,4,6,3] => ? = 1
Description
The number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000068The number of minimal elements in a poset. St000407The number of occurrences of the pattern 2143 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001344The neighbouring number of a permutation. St001330The hat guessing number of a graph. St001162The minimum jump of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St001715The number of non-records in a permutation. St001867The number of alignments of type EN of a signed permutation. St000367The number of simsun double descents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000516The number of stretching pairs of a permutation. St000629The defect of a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001301The first Betti number of the order complex associated with the poset. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001520The number of strict 3-descents. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001728The number of invisible descents of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000335The difference of lower and upper interactions. St000570The Edelman-Greene number of a permutation. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000908The length of the shortest maximal antichain in a poset. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000768The number of peaks in an integer composition. St001396Number of triples of incomparable elements in a finite poset. St000011The number of touch points (or returns) of a Dyck path. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001722The number of minimal chains with small intervals between a binary word and the top element. St000352The Elizalde-Pak rank of a permutation. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000054The first entry of the permutation. St000635The number of strictly order preserving maps of a poset into itself. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000842The breadth of a permutation. St001720The minimal length of a chain of small intervals in a lattice.
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