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Your data matches 120 different statistics following compositions of up to 3 maps.
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Matching statistic: St001604
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[-,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,5,-,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,1,-,3] => [2,5,1,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,2,1]
=> [2,1]
=> 0
[2,5,4,3,1] => [2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[3,1,5,+,2] => [3,1,5,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,1,5,-,2] => [3,1,5,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,+,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[3,-,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[3,+,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 0
[3,-,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 0
[3,+,5,+,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0
[3,-,5,+,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0
[3,+,5,-,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0
[3,-,5,-,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0
[3,5,1,+,2] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,5,1,-,2] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0
[3,5,2,1,4] => [3,5,2,1,4] => [2,2,1]
=> [2,1]
=> 0
[3,5,2,+,1] => [3,5,2,4,1] => [2,2,1]
=> [2,1]
=> 0
[3,5,2,-,1] => [3,5,2,4,1] => [2,2,1]
=> [2,1]
=> 0
[3,5,4,1,2] => [3,5,4,1,2] => [2,2,1]
=> [2,1]
=> 0
[3,5,4,2,1] => [3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[4,1,5,3,2] => [4,1,5,3,2] => [2,2,1]
=> [2,1]
=> 0
[4,+,1,5,3] => [4,2,1,5,3] => [2,2,1]
=> [2,1]
=> 0
[4,-,1,5,3] => [4,2,1,5,3] => [2,2,1]
=> [2,1]
=> 0
[4,+,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 0
[4,-,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 0
[4,+,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [2,1]
=> 0
[4,-,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [2,1]
=> 0
[4,3,1,5,2] => [4,3,1,5,2] => [2,2,1]
=> [2,1]
=> 0
[4,3,2,1,+] => [4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[4,3,2,1,-] => [4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[4,3,2,5,1] => [4,3,2,5,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[4,3,5,1,2] => [4,3,5,1,2] => [2,2,1]
=> [2,1]
=> 0
[4,3,5,2,1] => [4,3,5,2,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[4,5,1,3,2] => [4,5,1,3,2] => [2,2,1]
=> [2,1]
=> 0
[4,5,2,1,3] => [4,5,2,1,3] => [2,2,1]
=> [2,1]
=> 0
[4,5,2,3,1] => [4,5,2,3,1] => [2,2,1]
=> [2,1]
=> 0
[4,5,+,1,2] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0
[4,5,-,1,2] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0
[4,5,+,2,1] => [4,5,3,2,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[4,5,-,2,1] => [4,5,3,2,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[5,1,4,3,2] => [5,1,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[5,+,1,+,3] => [5,2,1,4,3] => [2,2,1]
=> [2,1]
=> 0
[5,-,1,+,3] => [5,2,1,4,3] => [2,2,1]
=> [2,1]
=> 0
[5,+,1,-,3] => [5,2,1,4,3] => [2,2,1]
=> [2,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001217
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[+,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[-,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,1,5,-,3] => [2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,5,1,-,3] => [2,5,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,5,4,3,1] => [2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[3,1,5,+,2] => [3,1,5,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,1,5,-,2] => [3,1,5,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,+,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,-,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,+,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,-,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,+,5,+,1] => [3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,-,5,+,1] => [3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,+,5,-,1] => [3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,-,5,-,1] => [3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,5,1,+,2] => [3,5,1,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,5,1,-,2] => [3,5,1,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,5,2,1,4] => [3,5,2,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,5,2,+,1] => [3,5,2,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,5,2,-,1] => [3,5,2,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,5,4,1,2] => [3,5,4,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,5,4,2,1] => [3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[4,1,5,3,2] => [4,1,5,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,+,1,5,3] => [4,2,1,5,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,-,1,5,3] => [4,2,1,5,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,+,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,-,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,+,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,-,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,3,1,5,2] => [4,3,1,5,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1,+] => [4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[4,3,2,1,-] => [4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[4,3,2,5,1] => [4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[4,3,5,1,2] => [4,3,5,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,3,5,2,1] => [4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[4,5,1,3,2] => [4,5,1,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,5,2,1,3] => [4,5,2,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,5,2,3,1] => [4,5,2,3,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,5,+,1,2] => [4,5,3,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,5,-,1,2] => [4,5,3,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,5,+,2,1] => [4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[4,5,-,2,1] => [4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[5,1,4,3,2] => [5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[5,+,1,+,3] => [5,2,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[5,-,1,+,3] => [5,2,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[5,+,1,-,3] => [5,2,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St001481
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[+,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[-,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,5,+,3] => [2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,5,-,3] => [2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,5,1,+,3] => [2,5,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,5,1,-,3] => [2,5,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,5,4,1,3] => [2,5,4,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,5,4,3,1] => [2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,1,5,+,2] => [3,1,5,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,5,-,2] => [3,1,5,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,+,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,-,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,+,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,-,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,+,5,+,1] => [3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,-,5,+,1] => [3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,+,5,-,1] => [3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,-,5,-,1] => [3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,5,1,+,2] => [3,5,1,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,5,1,-,2] => [3,5,1,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,5,2,1,4] => [3,5,2,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,5,2,+,1] => [3,5,2,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,5,2,-,1] => [3,5,2,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,5,4,1,2] => [3,5,4,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,5,4,2,1] => [3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,1,5,3,2] => [4,1,5,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,+,1,5,3] => [4,2,1,5,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,-,1,5,3] => [4,2,1,5,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,+,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,-,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,+,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,-,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,3,1,5,2] => [4,3,1,5,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,3,2,1,+] => [4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,3,2,1,-] => [4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,3,2,5,1] => [4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,3,5,1,2] => [4,3,5,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,3,5,2,1] => [4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,5,1,3,2] => [4,5,1,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,5,2,1,3] => [4,5,2,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,5,2,3,1] => [4,5,2,3,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,5,+,1,2] => [4,5,3,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,5,-,1,2] => [4,5,3,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,5,+,2,1] => [4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,5,-,2,1] => [4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[5,1,4,3,2] => [5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[5,+,1,+,3] => [5,2,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[5,-,1,+,3] => [5,2,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[5,+,1,-,3] => [5,2,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
Description
The minimal height of a peak of a Dyck path.
Matching statistic: St001526
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[+,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[-,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[2,1,5,+,3] => [2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[2,1,5,-,3] => [2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[2,5,1,+,3] => [2,5,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[2,5,1,-,3] => [2,5,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[2,5,4,1,3] => [2,5,4,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[2,5,4,3,1] => [2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[3,1,5,+,2] => [3,1,5,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,1,5,-,2] => [3,1,5,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,+,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,-,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,+,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,-,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,+,5,+,1] => [3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,-,5,+,1] => [3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,+,5,-,1] => [3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,-,5,-,1] => [3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,5,1,+,2] => [3,5,1,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,5,1,-,2] => [3,5,1,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,5,2,1,4] => [3,5,2,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,5,2,+,1] => [3,5,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,5,2,-,1] => [3,5,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,5,4,1,2] => [3,5,4,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,5,4,2,1] => [3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[4,1,5,3,2] => [4,1,5,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,+,1,5,3] => [4,2,1,5,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,-,1,5,3] => [4,2,1,5,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,+,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,-,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,+,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,-,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,3,1,5,2] => [4,3,1,5,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,3,2,1,+] => [4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[4,3,2,1,-] => [4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[4,3,2,5,1] => [4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[4,3,5,1,2] => [4,3,5,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,3,5,2,1] => [4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[4,5,1,3,2] => [4,5,1,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,5,2,1,3] => [4,5,2,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,5,2,3,1] => [4,5,2,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,5,+,1,2] => [4,5,3,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,5,-,1,2] => [4,5,3,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,5,+,2,1] => [4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[4,5,-,2,1] => [4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[5,1,4,3,2] => [5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[5,+,1,+,3] => [5,2,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[5,-,1,+,3] => [5,2,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[5,+,1,-,3] => [5,2,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[+,+,6,5,4,3] => [1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,+,6,5,4,3] => [1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,-,6,5,4,3] => [1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,-,6,5,4,3] => [1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,3,6,5,4,2] => [1,3,6,5,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,3,6,5,4,2] => [1,3,6,5,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,4,6,5,3,2] => [1,4,6,5,3,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,4,6,5,3,2] => [1,4,6,5,3,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,5,4,3,2,+] => [1,5,4,3,2,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,5,4,3,2,+] => [1,5,4,3,2,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,5,4,3,2,-] => [1,5,4,3,2,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,5,4,3,2,-] => [1,5,4,3,2,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,5,4,3,6,2] => [1,5,4,3,6,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,5,4,3,6,2] => [1,5,4,3,6,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,5,4,6,3,2] => [1,5,4,6,3,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,5,4,6,3,2] => [1,5,4,6,3,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,5,6,+,3,2] => [1,5,6,4,3,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,5,6,+,3,2] => [1,5,6,4,3,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,5,6,-,3,2] => [1,5,6,4,3,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,5,6,-,3,2] => [1,5,6,4,3,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,2,5,4,3] => [1,6,2,5,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,2,5,4,3] => [1,6,2,5,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,+,5,4,2] => [1,6,3,5,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,+,5,4,2] => [1,6,3,5,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,-,5,4,2] => [1,6,3,5,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,-,5,4,2] => [1,6,3,5,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,4,3,2,5] => [1,6,4,3,2,5] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,4,3,2,5] => [1,6,4,3,2,5] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,4,3,+,2] => [1,6,4,3,5,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,4,3,+,2] => [1,6,4,3,5,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,4,3,-,2] => [1,6,4,3,5,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,4,3,-,2] => [1,6,4,3,5,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,4,5,3,2] => [1,6,4,5,3,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,4,5,3,2] => [1,6,4,5,3,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,5,2,4,3] => [1,6,5,2,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,5,2,4,3] => [1,6,5,2,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,5,3,2,4] => [1,6,5,3,2,4] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,5,3,2,4] => [1,6,5,3,2,4] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,5,3,4,2] => [1,6,5,3,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,5,3,4,2] => [1,6,5,3,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,5,+,2,3] => [1,6,5,4,2,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,5,+,2,3] => [1,6,5,4,2,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,5,-,2,3] => [1,6,5,4,2,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,5,-,2,3] => [1,6,5,4,2,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,5,+,3,2] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,5,+,3,2] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[+,6,5,-,3,2] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[-,6,5,-,3,2] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[2,3,6,5,4,1] => [2,3,6,5,4,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[2,4,6,5,3,1] => [2,4,6,5,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 61%●distinct values known / distinct values provided: 50%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 61%●distinct values known / distinct values provided: 50%
Values
[4,3,2,1] => [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)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[+,5,4,3,2] => [1,5,4,3,2] => ([(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)
=> 1 = 0 + 1
[-,5,4,3,2] => [1,5,4,3,2] => ([(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)
=> 1 = 0 + 1
[2,1,5,+,3] => [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)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,5,-,3] => [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)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,5,1,+,3] => [2,5,1,4,3] => ([(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 + 1
[2,5,1,-,3] => [2,5,1,4,3] => ([(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 + 1
[2,5,4,1,3] => [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 + 1
[2,5,4,3,1] => [2,5,4,3,1] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,5,+,2] => [3,1,5,4,2] => ([(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 + 1
[3,1,5,-,2] => [3,1,5,4,2] => ([(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 + 1
[3,+,1,5,4] => [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)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,-,1,5,4] => [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)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,+,5,1,4] => [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)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,-,5,1,4] => [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)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,+,5,+,1] => [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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,-,5,+,1] => [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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,+,5,-,1] => [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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,-,5,-,1] => [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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,5,1,+,2] => [3,5,1,4,2] => ([(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 + 1
[3,5,1,-,2] => [3,5,1,4,2] => ([(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 + 1
[3,5,2,1,4] => [3,5,2,1,4] => ([(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 + 1
[3,5,2,+,1] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,5,2,-,1] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,5,4,1,2] => [3,5,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,5,4,2,1] => [3,5,4,2,1] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[4,1,5,3,2] => [4,1,5,3,2] => ([(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 + 1
[4,+,1,5,3] => [4,2,1,5,3] => ([(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 + 1
[4,-,1,5,3] => [4,2,1,5,3] => ([(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 + 1
[4,+,5,1,3] => [4,2,5,1,3] => ([(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 + 1
[4,-,5,1,3] => [4,2,5,1,3] => ([(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 + 1
[4,+,5,3,1] => [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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[4,-,5,3,1] => [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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[4,3,1,5,2] => [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 + 1
[4,3,2,1,+] => [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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[4,3,2,1,-] => [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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[4,3,2,5,1] => [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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[4,3,5,1,2] => [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,3,5,2,1] => [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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[4,5,1,3,2] => [4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,5,2,1,3] => [4,5,2,1,3] => ([(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)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,5,2,3,1] => [4,5,2,3,1] => ([(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)
=> 1 = 0 + 1
[4,5,+,1,2] => [4,5,3,1,2] => ([(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)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,5,-,1,2] => [4,5,3,1,2] => ([(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)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,5,+,2,1] => [4,5,3,2,1] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[4,5,-,2,1] => [4,5,3,2,1] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,1,4,3,2] => [5,1,4,3,2] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,+,1,+,3] => [5,2,1,4,3] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,-,1,+,3] => [5,2,1,4,3] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,+,1,-,3] => [5,2,1,4,3] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,-,1,-,3] => [5,2,1,4,3] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,+,4,1,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,-,4,1,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,+,4,3,1] => [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)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[5,-,4,3,1] => [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)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[5,3,1,+,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,3,1,-,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,3,2,1,4] => [5,3,2,1,4] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,3,2,+,1] => [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)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[5,3,2,-,1] => [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)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[5,3,4,1,2] => [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)
=> 1 = 0 + 1
[5,3,4,2,1] => [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)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[5,4,1,3,2] => [5,4,1,3,2] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,4,2,1,3] => [5,4,2,1,3] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,4,2,3,1] => [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)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[5,4,+,1,2] => [5,4,3,1,2] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,4,-,1,2] => [5,4,3,1,2] => ([(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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[5,4,+,2,1] => [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)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[5,4,-,2,1] => [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)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[+,+,6,5,4,3] => [1,2,6,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,+,6,5,4,3] => [1,2,6,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,-,6,5,4,3] => [1,2,6,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,-,6,5,4,3] => [1,2,6,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,3,2,6,+,4] => [1,3,2,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,3,2,6,+,4] => [1,3,2,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,3,2,6,-,4] => [1,3,2,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,10),(4,13),(5,13),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,+,6,+,4] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,-,6,+,4] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,+,6,-,4] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,-,6,-,4] => [2,1,3,6,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,9),(5,10),(6,11),(6,12),(8,9),(9,7),(10,7),(11,8),(12,8),(13,10)],14)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,6,3,5] => [2,1,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,6,+,3] => [2,1,4,6,5,3] => ([(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)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,4,6,-,3] => [2,1,4,6,5,3] => ([(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)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,5,3,6,4] => [2,1,5,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,5,+,6,3] => [2,1,5,4,6,3] => ([(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)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,5,-,6,3] => [2,1,5,4,6,3] => ([(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)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,5,6,3,4] => [2,1,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,5,6,4,3] => [2,1,5,6,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,9),(5,9),(6,7),(6,10),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,6,3,+,4] => [2,1,6,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,6,3,-,4] => [2,1,6,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,10),(10,11)],12)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,6,+,3,5] => [2,1,6,4,3,5] => ([(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)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,6,-,3,5] => [2,1,6,4,3,5] => ([(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)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,6,+,+,3] => [2,1,6,4,5,3] => ([(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),(8,10),(9,12),(10,12),(12,11)],13)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,6,-,+,3] => [2,1,6,4,5,3] => ([(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),(8,10),(9,12),(10,12),(12,11)],13)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,6,+,-,3] => [2,1,6,4,5,3] => ([(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),(8,10),(9,12),(10,12),(12,11)],13)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,6,-,-,3] => [2,1,6,4,5,3] => ([(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),(8,10),(9,12),(10,12),(12,11)],13)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,6,5,3,4] => [2,1,6,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,9),(5,9),(6,7),(6,10),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,6,5,4,3] => [2,1,6,5,4,3] => ([(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)
=> ? = 0 + 1
[2,3,1,6,+,4] => [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)
=> ([(0,1)],2)
=> ? = 0 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001771
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[4,3,2,1] => [4,3,2,1] => [3,2,1] => [3,2,1] => 0
[+,5,4,3,2] => [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,5,4,3,2] => [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,5,1,-,3] => [2,5,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[2,5,4,3,1] => [2,5,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
[3,1,5,+,2] => [3,1,5,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,1,5,-,2] => [3,1,5,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,+,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,5,1,4] => [3,2,5,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,5,1,4] => [3,2,5,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,5,+,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,-,5,+,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,+,5,-,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,-,5,-,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,5,1,+,2] => [3,5,1,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,5,1,-,2] => [3,5,1,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,5,2,1,4] => [3,5,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,5,2,+,1] => [3,5,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,5,2,-,1] => [3,5,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,5,4,1,2] => [3,5,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
[3,5,4,2,1] => [3,5,4,2,1] => [3,4,2,1] => [3,4,2,1] => 0
[4,1,5,3,2] => [4,1,5,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,1,5,3] => [4,2,1,5,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,-,1,5,3] => [4,2,1,5,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,+,5,1,3] => [4,2,5,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,-,5,1,3] => [4,2,5,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,+,5,3,1] => [4,2,5,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,-,5,3,1] => [4,2,5,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,3,1,5,2] => [4,3,1,5,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1,+] => [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,2,1,-] => [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,2,5,1] => [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,5,1,2] => [4,3,5,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,3,5,2,1] => [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,5,1,3,2] => [4,5,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,5,2,1,3] => [4,5,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,5,2,3,1] => [4,5,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,5,+,1,2] => [4,5,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,5,-,1,2] => [4,5,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,5,+,2,1] => [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,5,-,2,1] => [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[5,1,4,3,2] => [5,1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[5,+,1,+,3] => [5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[5,-,1,+,3] => [5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[5,+,1,-,3] => [5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,+,6,+,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,-,6,+,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,+,6,-,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,-,6,-,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,4,6,3,5] => [2,1,4,6,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,4,6,+,3] => [2,1,4,6,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,4,6,-,3] => [2,1,4,6,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,5,3,6,4] => [2,1,5,3,6,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
[2,1,5,+,3,+] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,-,3,+] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,+,3,-] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,-,3,-] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,+,6,3] => [2,1,5,4,6,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,-,6,3] => [2,1,5,4,6,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,6,3,4] => [2,1,5,6,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
[2,1,5,6,4,3] => [2,1,5,6,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,6,3,+,4] => [2,1,6,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,6,3,-,4] => [2,1,6,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,6,+,3,5] => [2,1,6,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,6,-,3,5] => [2,1,6,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,6,+,+,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,-,+,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,+,-,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,-,-,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,5,3,4] => [2,1,6,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[2,1,6,5,4,3] => [2,1,6,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,3,1,6,+,4] => [2,3,1,6,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,1,6,-,4] => [2,3,1,6,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,6,1,+,4] => [2,3,6,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,6,1,-,4] => [2,3,6,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,6,5,1,4] => [2,3,6,5,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[2,3,6,5,4,1] => [2,3,6,5,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
[2,4,1,3,6,5] => [2,4,1,3,6,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[2,4,1,6,3,5] => [2,4,1,6,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[2,4,1,6,+,3] => [2,4,1,6,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,1,6,-,3] => [2,4,1,6,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,+,1,6,5] => [2,4,3,1,6,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,-,1,6,5] => [2,4,3,1,6,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,+,6,1,5] => [2,4,3,6,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,-,6,1,5] => [2,4,3,6,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,+,6,+,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,-,6,+,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,+,6,-,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,-,6,-,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,6,1,3,5] => [2,4,6,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[2,4,6,1,+,3] => [2,4,6,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,6,1,-,3] => [2,4,6,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,6,3,1,5] => [2,4,6,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,6,3,+,1] => [2,4,6,3,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001870
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[4,3,2,1] => [4,3,2,1] => [3,2,1] => [3,2,1] => 0
[+,5,4,3,2] => [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,5,4,3,2] => [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,5,1,-,3] => [2,5,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[2,5,4,3,1] => [2,5,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
[3,1,5,+,2] => [3,1,5,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,1,5,-,2] => [3,1,5,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,+,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,5,1,4] => [3,2,5,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,5,1,4] => [3,2,5,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,5,+,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,-,5,+,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,+,5,-,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,-,5,-,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,5,1,+,2] => [3,5,1,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,5,1,-,2] => [3,5,1,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,5,2,1,4] => [3,5,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,5,2,+,1] => [3,5,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,5,2,-,1] => [3,5,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,5,4,1,2] => [3,5,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
[3,5,4,2,1] => [3,5,4,2,1] => [3,4,2,1] => [3,4,2,1] => 0
[4,1,5,3,2] => [4,1,5,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,1,5,3] => [4,2,1,5,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,-,1,5,3] => [4,2,1,5,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,+,5,1,3] => [4,2,5,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,-,5,1,3] => [4,2,5,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,+,5,3,1] => [4,2,5,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,-,5,3,1] => [4,2,5,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,3,1,5,2] => [4,3,1,5,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1,+] => [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,2,1,-] => [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,2,5,1] => [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,5,1,2] => [4,3,5,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,3,5,2,1] => [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,5,1,3,2] => [4,5,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,5,2,1,3] => [4,5,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,5,2,3,1] => [4,5,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,5,+,1,2] => [4,5,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,5,-,1,2] => [4,5,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,5,+,2,1] => [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,5,-,2,1] => [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[5,1,4,3,2] => [5,1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[5,+,1,+,3] => [5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[5,-,1,+,3] => [5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[5,+,1,-,3] => [5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,+,6,+,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,-,6,+,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,+,6,-,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,-,6,-,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,4,6,3,5] => [2,1,4,6,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,4,6,+,3] => [2,1,4,6,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,4,6,-,3] => [2,1,4,6,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,5,3,6,4] => [2,1,5,3,6,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
[2,1,5,+,3,+] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,-,3,+] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,+,3,-] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,-,3,-] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,+,6,3] => [2,1,5,4,6,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,-,6,3] => [2,1,5,4,6,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,6,3,4] => [2,1,5,6,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
[2,1,5,6,4,3] => [2,1,5,6,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,6,3,+,4] => [2,1,6,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,6,3,-,4] => [2,1,6,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,6,+,3,5] => [2,1,6,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,6,-,3,5] => [2,1,6,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,6,+,+,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,-,+,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,+,-,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,-,-,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,5,3,4] => [2,1,6,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[2,1,6,5,4,3] => [2,1,6,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,3,1,6,+,4] => [2,3,1,6,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,1,6,-,4] => [2,3,1,6,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,6,1,+,4] => [2,3,6,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,6,1,-,4] => [2,3,6,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,6,5,1,4] => [2,3,6,5,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[2,3,6,5,4,1] => [2,3,6,5,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
[2,4,1,3,6,5] => [2,4,1,3,6,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[2,4,1,6,3,5] => [2,4,1,6,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[2,4,1,6,+,3] => [2,4,1,6,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,1,6,-,3] => [2,4,1,6,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,+,1,6,5] => [2,4,3,1,6,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,-,1,6,5] => [2,4,3,1,6,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,+,6,1,5] => [2,4,3,6,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,-,6,1,5] => [2,4,3,6,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,+,6,+,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,-,6,+,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,+,6,-,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,-,6,-,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,6,1,3,5] => [2,4,6,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[2,4,6,1,+,3] => [2,4,6,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,6,1,-,3] => [2,4,6,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,6,3,1,5] => [2,4,6,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,6,3,+,1] => [2,4,6,3,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Matching statistic: St001895
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00252: Permutations —restriction⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[4,3,2,1] => [4,3,2,1] => [3,2,1] => [3,2,1] => 0
[+,5,4,3,2] => [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,5,4,3,2] => [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,5,1,-,3] => [2,5,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,5,4,1,3] => [2,5,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[2,5,4,3,1] => [2,5,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
[3,1,5,+,2] => [3,1,5,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,1,5,-,2] => [3,1,5,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,+,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,5,1,4] => [3,2,5,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,5,1,4] => [3,2,5,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,5,+,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,-,5,+,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,+,5,-,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,-,5,-,1] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,5,1,+,2] => [3,5,1,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,5,1,-,2] => [3,5,1,4,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,5,2,1,4] => [3,5,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,5,2,+,1] => [3,5,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,5,2,-,1] => [3,5,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[3,5,4,1,2] => [3,5,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
[3,5,4,2,1] => [3,5,4,2,1] => [3,4,2,1] => [3,4,2,1] => 0
[4,1,5,3,2] => [4,1,5,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,1,5,3] => [4,2,1,5,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,-,1,5,3] => [4,2,1,5,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,+,5,1,3] => [4,2,5,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,-,5,1,3] => [4,2,5,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,+,5,3,1] => [4,2,5,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,-,5,3,1] => [4,2,5,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,3,1,5,2] => [4,3,1,5,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1,+] => [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,2,1,-] => [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,2,5,1] => [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,5,1,2] => [4,3,5,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,3,5,2,1] => [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,5,1,3,2] => [4,5,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,5,2,1,3] => [4,5,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,5,2,3,1] => [4,5,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,5,+,1,2] => [4,5,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,5,-,1,2] => [4,5,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,5,+,2,1] => [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,5,-,2,1] => [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[5,1,4,3,2] => [5,1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[5,+,1,+,3] => [5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[5,-,1,+,3] => [5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[5,+,1,-,3] => [5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,+,6,+,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,-,6,+,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,+,6,-,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,-,6,-,4] => [2,1,3,6,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,4,6,3,5] => [2,1,4,6,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,4,6,+,3] => [2,1,4,6,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,4,6,-,3] => [2,1,4,6,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,5,3,6,4] => [2,1,5,3,6,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
[2,1,5,+,3,+] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,-,3,+] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,+,3,-] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,-,3,-] => [2,1,5,4,3,6] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,+,6,3] => [2,1,5,4,6,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,-,6,3] => [2,1,5,4,6,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,5,6,3,4] => [2,1,5,6,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
[2,1,5,6,4,3] => [2,1,5,6,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,1,6,3,+,4] => [2,1,6,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,6,3,-,4] => [2,1,6,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,6,+,3,5] => [2,1,6,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,6,-,3,5] => [2,1,6,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,6,+,+,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,-,+,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,+,-,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,-,-,3] => [2,1,6,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,6,5,3,4] => [2,1,6,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[2,1,6,5,4,3] => [2,1,6,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[2,3,1,6,+,4] => [2,3,1,6,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,1,6,-,4] => [2,3,1,6,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,6,1,+,4] => [2,3,6,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,6,1,-,4] => [2,3,6,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,6,5,1,4] => [2,3,6,5,1,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[2,3,6,5,4,1] => [2,3,6,5,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
[2,4,1,3,6,5] => [2,4,1,3,6,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[2,4,1,6,3,5] => [2,4,1,6,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[2,4,1,6,+,3] => [2,4,1,6,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,1,6,-,3] => [2,4,1,6,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,+,1,6,5] => [2,4,3,1,6,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,-,1,6,5] => [2,4,3,1,6,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,+,6,1,5] => [2,4,3,6,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,-,6,1,5] => [2,4,3,6,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,+,6,+,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,-,6,+,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,+,6,-,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,-,6,-,1] => [2,4,3,6,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,6,1,3,5] => [2,4,6,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[2,4,6,1,+,3] => [2,4,6,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,6,1,-,3] => [2,4,6,1,5,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,6,3,1,5] => [2,4,6,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,6,3,+,1] => [2,4,6,3,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
Description
The oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Matching statistic: St001260
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001260: Alternating sign matrices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001260: Alternating sign matrices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 1 = 0 + 1
[+,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[-,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[2,1,5,+,3] => [2,1,5,4,3] => [5,1,4,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[2,1,5,-,3] => [2,1,5,4,3] => [5,1,4,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[2,5,1,+,3] => [2,5,1,4,3] => [5,1,3,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[2,5,1,-,3] => [2,5,1,4,3] => [5,1,3,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[2,5,4,1,3] => [2,5,4,1,3] => [4,5,1,3,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[2,5,4,3,1] => [2,5,4,3,1] => [5,4,1,3,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[3,1,5,+,2] => [3,1,5,4,2] => [1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[3,1,5,-,2] => [3,1,5,4,2] => [1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[3,+,1,5,4] => [3,2,1,5,4] => [2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 1 = 0 + 1
[3,-,1,5,4] => [3,2,1,5,4] => [2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 1 = 0 + 1
[3,+,5,1,4] => [3,2,5,1,4] => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
[3,-,5,1,4] => [3,2,5,1,4] => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
[3,+,5,+,1] => [3,2,5,4,1] => [1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[3,-,5,+,1] => [3,2,5,4,1] => [1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[3,+,5,-,1] => [3,2,5,4,1] => [1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[3,-,5,-,1] => [3,2,5,4,1] => [1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[3,5,1,+,2] => [3,5,1,4,2] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[3,5,1,-,2] => [3,5,1,4,2] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[3,5,2,1,4] => [3,5,2,1,4] => [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
[3,5,2,+,1] => [3,5,2,4,1] => [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[3,5,2,-,1] => [3,5,2,4,1] => [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[3,5,4,1,2] => [3,5,4,1,2] => [1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[3,5,4,2,1] => [3,5,4,2,1] => [5,1,4,3,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[4,1,5,3,2] => [4,1,5,3,2] => [5,4,2,1,3] => [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[4,+,1,5,3] => [4,2,1,5,3] => [2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 1 = 0 + 1
[4,-,1,5,3] => [4,2,1,5,3] => [2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 1 = 0 + 1
[4,+,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
[4,-,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
[4,+,5,3,1] => [4,2,5,3,1] => [5,2,1,4,3] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[4,-,5,3,1] => [4,2,5,3,1] => [5,2,1,4,3] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[4,3,1,5,2] => [4,3,1,5,2] => [5,3,2,1,4] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 0 + 1
[4,3,2,1,+] => [4,3,2,1,5] => [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 0 + 1
[4,3,2,1,-] => [4,3,2,1,5] => [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 0 + 1
[4,3,2,5,1] => [4,3,2,5,1] => [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 0 + 1
[4,3,5,1,2] => [4,3,5,1,2] => [4,2,1,5,3] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
[4,3,5,2,1] => [4,3,5,2,1] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[4,5,1,3,2] => [4,5,1,3,2] => [5,3,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[4,5,2,1,3] => [4,5,2,1,3] => [4,3,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
[4,5,2,3,1] => [4,5,2,3,1] => [3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[4,5,+,1,2] => [4,5,3,1,2] => [4,1,5,3,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[4,5,-,1,2] => [4,5,3,1,2] => [4,1,5,3,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[4,5,+,2,1] => [4,5,3,2,1] => [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[4,5,-,2,1] => [4,5,3,2,1] => [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[5,1,4,3,2] => [5,1,4,3,2] => [5,4,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 0 + 1
[5,+,1,+,3] => [5,2,1,4,3] => [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[5,-,1,+,3] => [5,2,1,4,3] => [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[5,+,1,-,3] => [5,2,1,4,3] => [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[+,+,6,5,4,3] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[-,+,6,5,4,3] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[+,-,6,5,4,3] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[-,-,6,5,4,3] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[+,3,2,6,+,4] => [1,3,2,6,5,4] => [6,2,5,1,3,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[-,3,2,6,+,4] => [1,3,2,6,5,4] => [6,2,5,1,3,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[+,3,2,6,-,4] => [1,3,2,6,5,4] => [6,2,5,1,3,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[-,3,2,6,-,4] => [1,3,2,6,5,4] => [6,2,5,1,3,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[+,3,6,2,+,4] => [1,3,6,2,5,4] => [6,2,4,5,1,3] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[-,3,6,2,+,4] => [1,3,6,2,5,4] => [6,2,4,5,1,3] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[+,3,6,2,-,4] => [1,3,6,2,5,4] => [6,2,4,5,1,3] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[-,3,6,2,-,4] => [1,3,6,2,5,4] => [6,2,4,5,1,3] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[+,3,6,5,2,4] => [1,3,6,5,2,4] => [5,6,2,4,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[-,3,6,5,2,4] => [1,3,6,5,2,4] => [5,6,2,4,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[+,3,6,5,4,2] => [1,3,6,5,4,2] => [6,5,2,4,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[-,3,6,5,4,2] => [1,3,6,5,4,2] => [6,5,2,4,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[+,4,2,6,+,3] => [1,4,2,6,5,3] => [2,6,5,1,3,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[-,4,2,6,+,3] => [1,4,2,6,5,3] => [2,6,5,1,3,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[+,4,2,6,-,3] => [1,4,2,6,5,3] => [2,6,5,1,3,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[-,4,2,6,-,3] => [1,4,2,6,5,3] => [2,6,5,1,3,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[+,4,+,2,6,5] => [1,4,3,2,6,5] => [3,2,6,1,4,5] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0 + 1
[-,4,+,2,6,5] => [1,4,3,2,6,5] => [3,2,6,1,4,5] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0 + 1
[+,4,-,2,6,5] => [1,4,3,2,6,5] => [3,2,6,1,4,5] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0 + 1
[-,4,-,2,6,5] => [1,4,3,2,6,5] => [3,2,6,1,4,5] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0 + 1
[+,4,+,6,2,5] => [1,4,3,6,2,5] => [3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
[-,4,+,6,2,5] => [1,4,3,6,2,5] => [3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
[+,4,-,6,2,5] => [1,4,3,6,2,5] => [3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
[-,4,-,6,2,5] => [1,4,3,6,2,5] => [3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
[+,4,+,6,+,2] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[-,4,+,6,+,2] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[+,4,-,6,+,2] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[+,4,+,6,-,2] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[-,4,-,6,+,2] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[-,4,+,6,-,2] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[+,4,-,6,-,2] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[-,4,-,6,-,2] => [1,4,3,6,5,2] => [2,6,3,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[+,4,6,2,+,3] => [1,4,6,2,5,3] => [2,6,4,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[-,4,6,2,+,3] => [1,4,6,2,5,3] => [2,6,4,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[+,4,6,2,-,3] => [1,4,6,2,5,3] => [2,6,4,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[-,4,6,2,-,3] => [1,4,6,2,5,3] => [2,6,4,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0 + 1
[+,4,6,3,2,5] => [1,4,6,3,2,5] => [2,5,4,6,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
[-,4,6,3,2,5] => [1,4,6,3,2,5] => [2,5,4,6,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 1
[+,4,6,3,+,2] => [1,4,6,3,5,2] => [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
[-,4,6,3,+,2] => [1,4,6,3,5,2] => [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
[+,4,6,3,-,2] => [1,4,6,3,5,2] => [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
[-,4,6,3,-,2] => [1,4,6,3,5,2] => [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
[+,4,6,5,2,3] => [1,4,6,5,2,3] => [2,5,6,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
[-,4,6,5,2,3] => [1,4,6,5,2,3] => [2,5,6,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 1
[+,4,6,5,3,2] => [1,4,6,5,3,2] => [6,2,5,4,1,3] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
[-,4,6,5,3,2] => [1,4,6,5,3,2] => [6,2,5,4,1,3] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 0 + 1
Description
The permanent of an alternating sign matrix.
Matching statistic: St001198
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Values
[4,3,2,1] => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[+,5,4,3,2] => [1,4,5,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[-,5,4,3,2] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,1,5,+,3] => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,1,5,-,3] => [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,5,1,+,3] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,5,1,-,3] => [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,5,4,1,3] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[2,5,4,3,1] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,1,5,+,2] => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,1,5,-,2] => [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,+,1,5,4] => [2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,-,1,5,4] => [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,+,5,1,4] => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,-,5,1,4] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,+,5,+,1] => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,-,5,+,1] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,+,5,-,1] => [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,-,5,-,1] => [5,1,2,3,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,5,1,+,2] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,5,1,-,2] => [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,5,2,1,4] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,5,2,+,1] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,5,2,-,1] => [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,5,4,1,2] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,5,4,2,1] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,1,5,3,2] => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,+,1,5,3] => [2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,-,1,5,3] => [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,+,5,1,3] => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,-,5,1,3] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,+,5,3,1] => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,-,5,3,1] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,3,1,5,2] => [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,3,2,1,+] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,3,2,1,-] => [3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[4,3,2,5,1] => [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,3,5,1,2] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,3,5,2,1] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,5,1,3,2] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,5,2,1,3] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,5,2,3,1] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,5,+,1,2] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,5,-,1,2] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,5,+,2,1] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[4,5,-,2,1] => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,1,4,3,2] => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,+,1,+,3] => [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,-,1,+,3] => [3,4,5,1,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,+,1,-,3] => [2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,-,1,-,3] => [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,+,4,1,3] => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[+,+,6,5,4,3] => [1,2,5,6,3,4] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[+,-,6,5,4,3] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,3,2,6,+,4] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,3,2,6,-,4] => [1,3,6,2,4,5] => [1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,3,6,2,+,4] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,3,6,2,-,4] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,3,6,5,2,4] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,3,6,5,4,2] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,2,6,+,3] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,2,6,-,3] => [1,3,6,2,4,5] => [1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,+,2,6,5] => [1,3,4,6,2,5] => [1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,-,2,6,5] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,+,6,2,5] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,-,6,2,5] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,+,6,+,2] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,-,6,+,2] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,+,6,-,2] => [1,3,6,2,4,5] => [1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,-,6,-,2] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,6,2,+,3] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,6,2,-,3] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,6,3,2,5] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,6,3,+,2] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,6,3,-,2] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,6,5,2,3] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,4,6,5,3,2] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,2,6,4,3] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,+,2,6,4] => [1,3,4,6,2,5] => [1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,-,2,6,4] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,+,6,2,4] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,-,6,2,4] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,+,6,4,2] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,-,6,4,2] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,4,2,6,3] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,4,3,2,+] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,4,3,2,-] => [1,4,5,2,3,6] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[-,5,4,3,2,-] => [4,5,1,2,3,6] => [5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[+,5,4,3,6,2] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,4,6,2,3] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,4,6,3,2] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,6,2,4,3] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,6,3,2,4] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,6,3,4,2] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,6,+,2,3] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,6,-,2,3] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,6,+,3,2] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,5,6,-,3,2] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,6,2,5,4,3] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[+,6,+,2,+,4] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 110 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St001208The 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 $K[x]/(x^n)$. St001429The number of negative entries in a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001890The maximum magnitude of the Möbius function of a poset. St001569The maximal modular displacement of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000221The number of strong fixed points of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000750The number of occurrences of the pattern 4213 in a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001386The number of prime labellings of a graph. St001430The number of positive entries in a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000056The decomposition (or block) number of a permutation. St000069The number of maximal elements of a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000486The number of cycles of length at least 3 of a permutation. St000545The number of parabolic double cosets with minimal element being the given permutation. St000694The number of affine bounded permutations that project to a given permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001081The number of minimal length factorizations of a permutation into star transpositions. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001344The neighbouring number of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001555The order of a signed permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001893The flag descent of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001851The number of Hecke atoms of a signed permutation. St001896The number of right descents of a signed permutations. St001812The biclique partition number of a graph. St000068The number of minimal elements in a poset.
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