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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000260
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 0
[1,2] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001597
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> []
=> [[],[]]
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> [[1],[]]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[4,3],[]]
=> 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[4,3],[]]
=> 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 1
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[4,3,3,2,1],[]]
=> ? = 3
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[3,3,3,2,1],[]]
=> ? = 2
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[3,3,3,2,1],[]]
=> ? = 2
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[4,2,2,2,1],[]]
=> ? = 2
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[4,2,2,2,1],[]]
=> ? = 2
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[3,2,2,2,1],[]]
=> ? = 2
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[3,2,2,2,1],[]]
=> ? = 2
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 2
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 2
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 2
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 2
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 2
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 2
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[4,4,3,1,1],[]]
=> ? = 2
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[4,3,3,1,1],[]]
=> ? = 2
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[3,3,3,1,1],[]]
=> ? = 2
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[3,3,3,1,1],[]]
=> ? = 2
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> ? = 2
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> ? = 2
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 2
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 2
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 2
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 2
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 2
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 2
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[4,4,1,1,1],[]]
=> ? = 2
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? = 2
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? = 2
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[4,4,1,1,1],[]]
=> ? = 2
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? = 2
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? = 2
[1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 2
[1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 2
[1,4,6,3,2,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 2
[1,4,6,3,5,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 2
[1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 2
[1,4,6,5,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 2
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[5,3,3,2],[]]
=> ? = 2
[2,1,3,5,6,4] => [1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[4,3,3,2],[]]
=> ? = 2
[2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[3,3,3,2],[]]
=> ? = 2
[2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[3,3,3,2],[]]
=> ? = 2
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[5,2,2,2],[]]
=> ? = 2
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[4,2,2,2],[]]
=> ? = 2
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[5,2,2,2],[]]
=> ? = 2
[2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[4,2,2,2],[]]
=> ? = 2
[2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [[3,2,2,2],[]]
=> ? = 2
[2,1,5,6,4,3] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [[3,2,2,2],[]]
=> ? = 2
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 2
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 2
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 2
[2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 2
Description
The Frobenius rank of a skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St001330
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 75%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 75%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 + 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 + 1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2 + 1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,3,5,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,3,5,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,4,2,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,4,3,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,4,5,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,5,2,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,5,3,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,5,3,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[5,1,3,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[5,1,4,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000264
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> ? = 0 + 1
[1,2] => ([],2)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 2 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([],2)
=> ? = 2 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 2 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 2 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 2 + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 + 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 + 1
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 2 + 1
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 + 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 + 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 + 1
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1 + 1
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 + 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? = 3 + 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],4)
=> ? = 2 + 1
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> ([],3)
=> ? = 2 + 1
[1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],3)
=> ? = 2 + 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],3)
=> ? = 2 + 1
[1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ([],3)
=> ? = 2 + 1
[1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,6,4,2,5] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,2,6,5,3] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,3,6,2,5] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[3,1,5,4,2,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[3,2,5,1,4,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[4,1,2,5,3,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[4,1,3,5,2,6] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[4,2,1,5,3,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,4,7,3,6,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,2,4,7,5,3,6] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,2,6,3,5,7,4] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,2,6,4,3,7,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,3,2,5,7,4,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,3,2,6,4,7,5] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,3,4,7,2,6,5] => ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,3,4,7,5,2,6] => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,3,6,2,7,4,5] => ([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,6,2,7,5,4] => ([(1,3),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,7,2,4,6,5] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,3,7,2,5,4,6] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,3,7,2,5,6,4] => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,7,2,6,4,5] => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,7,2,6,5,4] => ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,7,4,2,5,6] => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,3,7,4,2,6,5] => ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,7,4,5,2,6] => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,7,5,2,4,6] => ([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,7,5,4,2,6] => ([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,2,5,3,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,4,2,5,7,6,3] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,4,2,6,7,3,5] => ([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,2,6,7,5,3] => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,2,7,5,6,3] => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,2,7,6,3,5] => ([(1,3),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,2,7,6,5,3] => ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,3,5,7,2,6] => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,4,3,7,2,5,6] => ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,4,3,7,2,6,5] => ([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,3,7,5,2,6] => ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,3,7,6,4] => ([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,5,2,4,7,6,3] => ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,7,3,4,6] => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001269
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 75%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 75%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 1
[1,2,3,5,6,4] => [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]
=> [5,4,6,3,2,1,7] => ? = 3
[1,2,3,6,4,5] => [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]
=> [4,5,6,3,2,1,7] => ? = 2
[1,2,3,6,5,4] => [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]
=> [4,5,6,3,2,1,7] => ? = 2
[1,2,5,3,6,4] => [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]
=> [5,3,4,6,2,1,7] => ? = 2
[1,2,5,4,6,3] => [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]
=> [5,3,4,6,2,1,7] => ? = 2
[1,2,5,6,3,4] => [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]
=> [4,3,5,6,2,1,7] => ? = 2
[1,2,5,6,4,3] => [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]
=> [4,3,5,6,2,1,7] => ? = 2
[1,2,6,3,4,5] => [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]
=> [3,4,5,6,2,1,7] => ? = 2
[1,2,6,3,5,4] => [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]
=> [3,4,5,6,2,1,7] => ? = 2
[1,2,6,4,3,5] => [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]
=> [3,4,5,6,2,1,7] => ? = 2
[1,2,6,4,5,3] => [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]
=> [3,4,5,6,2,1,7] => ? = 2
[1,2,6,5,3,4] => [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]
=> [3,4,5,6,2,1,7] => ? = 2
[1,2,6,5,4,3] => [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]
=> [3,4,5,6,2,1,7] => ? = 2
[1,3,2,4,6,5] => [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]
=> [5,6,4,2,3,1,7] => ? = 2
[1,3,2,5,6,4] => [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]
=> [5,4,6,2,3,1,7] => ? = 2
[1,3,2,6,4,5] => [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]
=> [4,5,6,2,3,1,7] => ? = 2
[1,3,2,6,5,4] => [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]
=> [4,5,6,2,3,1,7] => ? = 2
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,1,7] => ? = 2
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,1,7] => ? = 2
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => ? = 2
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => ? = 2
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => ? = 2
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => ? = 2
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => ? = 2
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => ? = 2
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,1,7] => ? = 2
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => ? = 2
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => ? = 2
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,1,7] => ? = 2
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => ? = 2
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => ? = 2
[1,4,6,2,3,5] => [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]
=> [3,4,2,5,6,1,7] => ? = 2
[1,4,6,2,5,3] => [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]
=> [3,4,2,5,6,1,7] => ? = 2
[1,4,6,3,2,5] => [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]
=> [3,4,2,5,6,1,7] => ? = 2
[1,4,6,3,5,2] => [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]
=> [3,4,2,5,6,1,7] => ? = 2
[1,4,6,5,2,3] => [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]
=> [3,4,2,5,6,1,7] => ? = 2
[1,4,6,5,3,2] => [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]
=> [3,4,2,5,6,1,7] => ? = 2
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,3,5,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,3,5,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
Description
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
Matching statistic: St001874
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001874: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 75%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001874: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 75%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5] => 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,5,6,3] => 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,6,3] => 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,6,2,3,5] => 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,6,3,4] => 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,6,2,3,4] => 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,3,6,4] => 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,6,3,4] => 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,6,2,3,4] => 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,3,6,4] => 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => 1
[1,2,3,5,6,4] => [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]
=> [1,3,5,2,7,4,6] => ? = 3
[1,2,3,6,4,5] => [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]
=> [1,3,5,7,2,4,6] => ? = 2
[1,2,3,6,5,4] => [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]
=> [1,3,5,7,2,4,6] => ? = 2
[1,2,5,3,6,4] => [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]
=> [1,3,5,2,6,7,4] => ? = 2
[1,2,5,4,6,3] => [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]
=> [1,3,5,2,6,7,4] => ? = 2
[1,2,5,6,3,4] => [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]
=> [1,3,5,6,2,7,4] => ? = 2
[1,2,5,6,4,3] => [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]
=> [1,3,5,6,2,7,4] => ? = 2
[1,2,6,3,4,5] => [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]
=> [1,3,5,6,7,2,4] => ? = 2
[1,2,6,3,5,4] => [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]
=> [1,3,5,6,7,2,4] => ? = 2
[1,2,6,4,3,5] => [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]
=> [1,3,5,6,7,2,4] => ? = 2
[1,2,6,4,5,3] => [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]
=> [1,3,5,6,7,2,4] => ? = 2
[1,2,6,5,3,4] => [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]
=> [1,3,5,6,7,2,4] => ? = 2
[1,2,6,5,4,3] => [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]
=> [1,3,5,6,7,2,4] => ? = 2
[1,3,2,4,6,5] => [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]
=> [1,3,6,2,4,7,5] => ? = 2
[1,3,2,5,6,4] => [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]
=> [1,3,6,2,7,4,5] => ? = 2
[1,3,2,6,4,5] => [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]
=> [1,3,6,7,2,4,5] => ? = 2
[1,3,2,6,5,4] => [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]
=> [1,3,6,7,2,4,5] => ? = 2
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,4,6,2,7,5] => ? = 2
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,4,6,2,7,5] => ? = 2
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,4,6,7,2,5] => ? = 2
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,4,6,7,2,5] => ? = 2
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,4,6,7,2,5] => ? = 2
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,4,6,7,2,5] => ? = 2
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,4,6,7,2,5] => ? = 2
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,4,6,7,2,5] => ? = 2
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,3,7,2,4,5,6] => ? = 2
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => ? = 2
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => ? = 2
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,3,7,2,4,5,6] => ? = 2
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => ? = 2
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => ? = 2
[1,4,6,2,3,5] => [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]
=> [1,3,4,5,7,2,6] => ? = 2
[1,4,6,2,5,3] => [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]
=> [1,3,4,5,7,2,6] => ? = 2
[1,4,6,3,2,5] => [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]
=> [1,3,4,5,7,2,6] => ? = 2
[1,4,6,3,5,2] => [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]
=> [1,3,4,5,7,2,6] => ? = 2
[1,4,6,5,2,3] => [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]
=> [1,3,4,5,7,2,6] => ? = 2
[1,4,6,5,3,2] => [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]
=> [1,3,4,5,7,2,6] => ? = 2
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,3,5,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,3,5,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
[1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => ? = 1
Description
Lusztig's a-function for the symmetric group.
Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$
by the Robinson-Schensted correspondence and
$\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$
where $Q(x)'$ is the transposed tableau.
Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$.
See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.
Matching statistic: St001569
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [1] => ? = 0
[1,2] => [1,0,1,0]
=> [2,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ? = 3
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ? = 2
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ? = 2
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => ? = 2
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => ? = 2
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => ? = 2
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => ? = 2
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ? = 2
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ? = 2
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ? = 2
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ? = 2
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ? = 2
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ? = 2
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => ? = 2
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => ? = 2
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => ? = 2
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => ? = 2
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => ? = 2
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => ? = 2
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ? = 2
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ? = 2
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ? = 2
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ? = 2
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ? = 2
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ? = 2
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => ? = 2
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => ? = 2
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => ? = 2
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => ? = 2
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => ? = 2
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => ? = 2
[1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 2
[1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 2
[1,4,6,3,2,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 2
[1,4,6,3,5,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 2
[1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 2
[1,4,6,5,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 2
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,3,5,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,6,3,5,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
Description
The maximal modular displacement of a permutation.
This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Matching statistic: St001876
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0
[1,2] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,2] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1
[2,1,3] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2
[1,4,2,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([],1)
=> ? = 1
[1,4,3,2] => [2,1,3,4] => ([(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)
=> ([(0,1)],2)
=> ? = 1
[2,1,3,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2
[2,1,4,3] => [3,2,1,4] => ([(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)
=> ([(0,1)],2)
=> ? = 2
[3,1,2,4] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,4,5,3] => [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)
=> ([],1)
=> ? = 2
[1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 2
[1,2,5,4,3] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 2
[1,3,2,5,4] => [2,4,3,1,5] => ([(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)
=> ([(0,1)],2)
=> ? = 2
[1,3,5,2,4] => [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),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
[1,3,5,4,2] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2
[1,5,2,3,4] => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 1
[1,5,2,4,3] => [2,1,4,3,5] => ([(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)
=> ([(0,1)],2)
=> ? = 1
[1,5,3,2,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ([],1)
=> ? = 1
[1,5,3,4,2] => [2,1,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 1
[1,5,4,2,3] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ([],1)
=> ? = 1
[1,5,4,3,2] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ([(0,1)],2)
=> ? = 1
[2,1,4,3,5] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1)],2)
=> ? = 2
[2,1,4,5,3] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 2
[2,1,5,3,4] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ([],1)
=> ? = 2
[2,1,5,4,3] => [3,2,1,4,5] => ([(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)
=> ([(0,1)],2)
=> ? = 2
[2,3,1,4,5] => [3,4,2,5,1] => ([(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)
=> ([(0,1)],2)
=> ? = 2
[2,3,1,5,4] => [3,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1)],2)
=> ? = 2
[3,1,2,4,5] => [4,2,3,5,1] => ([(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)
=> ([(0,1)],2)
=> ? = 2
[3,1,2,5,4] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ([(0,1)],2)
=> ? = 2
[3,1,4,2,5] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2
[3,2,1,4,5] => [4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,1)],2)
=> ? = 2
[3,2,1,5,4] => [4,3,2,1,5] => ([(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)
=> ([(0,1)],2)
=> ? = 2
[3,2,4,1,5] => [4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 2
[4,1,2,3,5] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,1,3,2,5] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3,5] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1,5] => [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2,5] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,3,5,6,4] => [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)
=> ([],1)
=> ? = 3
[1,2,3,6,4,5] => [2,3,4,1,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ([],1)
=> ? = 2
[1,2,3,6,5,4] => [2,3,4,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ([(0,1)],2)
=> ? = 2
[1,2,5,3,6,4] => [2,3,6,4,1,5] => ([(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)
=> ([],1)
=> ? = 2
[1,2,5,4,6,3] => [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)
=> ([],1)
=> ? = 2
[1,2,5,6,3,4] => [2,3,6,1,5,4] => ([(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)
=> ([],1)
=> ? = 2
[1,2,5,6,4,3] => [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)
=> ([],1)
=> ? = 2
[1,2,6,3,4,5] => [2,3,1,5,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ([],1)
=> ? = 2
[1,2,6,3,5,4] => [2,3,1,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ([(0,1)],2)
=> ? = 2
[1,2,6,4,3,5] => [2,3,1,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,8),(3,7),(4,9),(5,9),(6,7),(6,8),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
=> ([],1)
=> ? = 2
[1,2,6,4,5,3] => [2,3,1,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ([],1)
=> ? = 2
[1,2,6,5,3,4] => [2,3,1,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,11),(6,9),(6,10),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> ([],1)
=> ? = 2
[1,2,6,5,4,3] => [2,3,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
=> ([(0,1)],2)
=> ? = 2
[1,3,2,4,6,5] => [2,4,3,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,10),(4,9),(5,8),(6,11),(7,9),(7,10),(9,12),(10,12),(11,8),(12,11)],13)
=> ([(0,1)],2)
=> ? = 2
[1,3,2,5,6,4] => [2,4,3,6,1,5] => ([(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)
=> ([],1)
=> ? = 2
[1,3,2,6,4,5] => [2,4,3,1,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ([],1)
=> ? = 2
[1,3,2,6,5,4] => [2,4,3,1,5,6] => ([(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),(10,9),(11,10)],13)
=> ([(0,1)],2)
=> ? = 2
[1,3,5,6,2,4] => [2,4,6,1,5,3] => ([(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),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,3,5,6,4,2] => [2,4,6,1,3,5] => ([(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),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[5,1,2,3,4,6] => [6,2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,1,2,4,3,6] => [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,1,3,2,4,6] => [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,1,3,4,2,6] => [6,2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,1,4,2,3,6] => [6,2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,1,4,3,2,6] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,2,1,3,4,6] => [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,2,1,4,3,6] => [6,3,2,5,4,1] => ([(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),(9,12),(10,12),(11,9),(11,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,2,3,1,4,6] => [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,2,3,4,1,6] => [6,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,2,4,1,3,6] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,2,4,3,1,6] => [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,3,1,2,4,6] => [6,4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,3,1,4,2,6] => [6,4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,3,2,1,4,6] => [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,3,2,4,1,6] => [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,3,4,1,2,6] => [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,3,4,2,1,6] => [6,4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,4,1,2,3,6] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,4,1,3,2,6] => [6,5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,4,2,1,3,6] => [6,5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,4,2,3,1,6] => [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[5,4,3,1,2,6] => [6,5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
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