Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St000740
(load all 3 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => [3,1,2] => 2
[1,3,2] => [1,2,3] => [2,3,1] => [3,1,2] => 2
[2,1,3] => [1,2,3] => [2,3,1] => [3,1,2] => 2
[2,3,1] => [1,2,3] => [2,3,1] => [3,1,2] => 2
[3,1,2] => [1,3,2] => [3,2,1] => [3,2,1] => 1
[3,2,1] => [1,3,2] => [3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 2
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 2
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 2
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 3
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 1
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 3
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 1
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 3
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 3
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 2
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 2
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 2
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => [5,1,2,4,3] => 3
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => [5,1,2,4,3] => 3
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => [5,1,2,4,3] => 3
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => [5,1,2,4,3] => 3
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [5,1,3,2,4] => 4
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => [5,1,3,4,2] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => [5,1,3,2,4] => 4
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => [5,1,3,4,2] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => [5,1,3,2,4] => 4
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Matching statistic: St001291
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
 => 1
[2,1] => [1,2] => [2,1] => [1,1,0,0]
 => 1
[1,2,3] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[1,3,2] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1,3] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,3,1] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[3,1,2] => [1,3,2] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[3,2,1] => [1,3,2] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => 3
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => 3
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => 3
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => 3
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => 4
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => 4
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => 4
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000427
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000427: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => [2,1] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[1,3,2] => [1,2,3] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[2,1,3] => [1,2,3] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[2,3,1] => [1,2,3] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[3,2,1] => [1,3,2] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 3 = 4 - 1
Description
The number of occurrences of the pattern 123 or of the pattern 231 in a permutation.
Matching statistic: St001084
(load all 2 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001084: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[3,2,1] => [1,3,2] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 3 - 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 3 - 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 3 - 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 3 - 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 3 - 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 3 - 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 3 - 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,2,4,3] => 2 = 3 - 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => [3,4,1,2] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,2,4,3] => 2 = 3 - 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => [3,4,1,2] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,2,4,3] => 2 = 3 - 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,2,4,3] => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [3,4,2,1] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [2,3,1,4] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [3,4,2,1] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [2,3,1,4] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [2,3,1,4] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [2,3,1,4] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 3 = 4 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => 2 = 3 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 3 = 4 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 3 = 4 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 3 = 4 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 3 = 4 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => 2 = 3 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => 2 = 3 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,5,4] => 3 = 4 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [3,4,5,1,2] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,5,4] => 3 = 4 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => [3,4,5,1,2] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,5,4] => 3 = 4 - 1
Description
The number of occurrences of the vincular pattern |1-23 in a permutation. This is the number of occurrences of the pattern $123$, where the first two matched entries are the first two entries of the permutation. In other words, this statistic is zero, if the first entry of the permutation is larger than the second, and it is the number of entries larger than the second entry otherwise.
Matching statistic: St000800
(load all 6 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000800: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => 1 = 2 - 1
[1,3,2] => [1,2,3] => [2,3,1] => 1 = 2 - 1
[2,1,3] => [1,2,3] => [2,3,1] => 1 = 2 - 1
[2,3,1] => [1,2,3] => [2,3,1] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [2,1,3] => 0 = 1 - 1
[3,2,1] => [1,3,2] => [2,1,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 2 = 3 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 2 = 3 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 2 = 3 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 2 = 3 - 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 2 = 3 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 2 = 3 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 2 = 3 - 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => 2 = 3 - 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => 2 = 3 - 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => 2 = 3 - 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 4 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 3 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 4 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 4 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 4 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 3 = 4 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 3 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 3 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 4 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 4 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 4 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 4 - 1
Description
The number of occurrences of the vincular pattern |231 in a permutation. This is the number of occurrences of the pattern $(2,3,1)$, such that the letter matched by $2$ is the first entry of the permutation.
Matching statistic: St000060
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000060: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 1
[1,2] => [1,2] => [2,1] => [1,2] => 1
[2,1] => [1,2] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [2,3,1] => [1,2,3] => 2
[1,3,2] => [1,2,3] => [2,3,1] => [1,2,3] => 2
[2,1,3] => [1,2,3] => [2,3,1] => [1,2,3] => 2
[2,3,1] => [1,2,3] => [2,3,1] => [1,2,3] => 2
[3,1,2] => [1,3,2] => [3,2,1] => [2,1,3] => 1
[3,2,1] => [1,3,2] => [3,2,1] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 2
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 2
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 3
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 2
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 3
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 1
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 3
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 1
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 3
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 3
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => [3,2,1,4] => 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 2
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => [3,2,1,4] => 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 2
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 2
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 3
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 3
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 4
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 3
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 3
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 4
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => [1,3,4,2,5] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 4
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => [1,3,4,2,5] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 4
[1,4,5,3,2] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 4
Description
The greater neighbor of the maximum. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the smallest path leaf label of the binary tree associated to a permutation ([[St000724]]), see also [3].
Matching statistic: St000799
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000799: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [2,1] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[1,3,2] => [1,2,3] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[2,1,3] => [1,2,3] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[2,3,1] => [1,2,3] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[3,2,1] => [1,3,2] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [3,2,4,1] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [3,2,4,1] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [3,2,4,1] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,2,4,1] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [3,4,2,1] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [3,1,2,4] => 2 = 3 - 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [3,4,2,1] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [3,1,2,4] => 2 = 3 - 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [3,1,2,4] => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [3,4,1,2] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [3,4,1,2] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 2 = 3 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 2 = 3 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 2 = 3 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [4,3,1,2,5] => 3 = 4 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [4,3,5,2,1] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [4,3,1,2,5] => 3 = 4 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [4,3,5,2,1] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [4,3,1,2,5] => 3 = 4 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => [4,3,1,2,5] => 3 = 4 - 1
Description
The number of occurrences of the vincular pattern |213 in a permutation. This is the number of occurrences of the pattern $(2,1,3)$, such that the letter matched by $2$ is the first entry of the permutation.
Matching statistic: St000219
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000219: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [2,1] => [2,1] => ? = 1 - 1
[2,1] => [1,2] => [2,1] => [2,1] => ? = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[1,3,2] => [1,2,3] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[2,1,3] => [1,2,3] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[2,3,1] => [1,2,3] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[3,2,1] => [1,3,2] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,5,2,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => [2,5,1,4,3] => 1 = 2 - 1
[1,5,2,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => [2,5,4,1,3] => 2 = 3 - 1
Description
The number of occurrences of the pattern 231 in a permutation.
Matching statistic: St000199
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [[1]]
 => 1
[1,2] => [1,2] => [2,1] => [[0,1],[1,0]]
 => 1
[2,1] => [1,2] => [2,1] => [[0,1],[1,0]]
 => 1
[1,2,3] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 2
[1,3,2] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 2
[2,1,3] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 2
[2,3,1] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 2
[3,1,2] => [1,3,2] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 1
[3,2,1] => [1,3,2] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 2
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 2
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 2
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 3
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 1
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 3
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 1
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 3
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 3
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 2
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 2
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 2
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 3
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 3
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 3
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 3
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 4
[1,4,2,5,3] => [1,2,4,5,3] => [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]]
 => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 4
[1,4,3,5,2] => [1,2,4,5,3] => [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]]
 => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 4
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 4
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,4,6,3,5] => [1,2,3,4,6,5] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 4
[1,2,4,6,5,3] => [1,2,3,4,6,5] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 4
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,5,3,6,4] => [1,2,3,5,6,4] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 3
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,5,4,6,3] => [1,2,3,5,6,4] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 3
[1,2,5,6,3,4] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,5,6,4,3] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 3
[1,2,6,3,5,4] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 4
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 3
[1,2,6,4,5,3] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 4
[1,2,6,5,3,4] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 4
[1,2,6,5,4,3] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 4
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,2,6,4,5] => [1,2,3,4,6,5] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 4
[1,3,2,6,5,4] => [1,2,3,4,6,5] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 4
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,4,6,2,5] => [1,2,3,4,6,5] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 4
[1,3,4,6,5,2] => [1,2,3,4,6,5] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 4
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,5,2,6,4] => [1,2,3,5,6,4] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 3
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,5,4,6,2] => [1,2,3,5,6,4] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 3
[1,3,5,6,2,4] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,5,6,4,2] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 3
[1,3,6,2,5,4] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 4
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 3
[1,3,6,4,5,2] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 4
[1,3,6,5,2,4] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 4
[1,3,6,5,4,2] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 4
[1,4,2,3,5,6] => [1,2,4,3,5,6] => [2,4,3,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [2,4,3,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 5
Description
The column of the unique '1' in the last row of the alternating sign matrix.
Matching statistic: St001557
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [2,1] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [1,3,2] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2 = 3 - 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2 = 3 - 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2 = 3 - 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2 = 3 - 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2 = 3 - 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2 = 3 - 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,4,2,3] => 2 = 3 - 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 3 - 1
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [1,2,4,3] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 3 - 1
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [1,2,4,3] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 3 - 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [1,3,4,2] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [1,3,4,2] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [1,3,4,2] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [1,3,4,2] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 3 = 4 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,4,2,5,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,4,2,5,3] => 2 = 3 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 3 = 4 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 3 = 4 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 3 = 4 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,2,4,3] => 3 = 4 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,4,2,5,3] => 2 = 3 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,4,2,5,3] => 2 = 3 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 3 = 4 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,3,4,2,5] => 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 3 = 4 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [1,3,4,2,5] => 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 3 = 4 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [5,3,4,2,1] => [1,5,2,3,4] => 3 = 4 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => [1,5,2,6,3,4] => ? = 4 - 1
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => [1,5,2,6,3,4] => ? = 4 - 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,2,4,6,3,5] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => [1,5,2,6,3,4] => ? = 4 - 1
[1,2,4,6,5,3] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => [1,5,2,6,3,4] => ? = 4 - 1
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => [1,6,2,4,3,5] => ? = 5 - 1
[1,2,5,3,6,4] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => [1,4,3,5,2,6] => ? = 3 - 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => [1,6,2,4,3,5] => ? = 5 - 1
[1,2,5,4,6,3] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => [1,4,3,5,2,6] => ? = 3 - 1
[1,2,5,6,3,4] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => [1,6,2,4,3,5] => ? = 5 - 1
[1,2,5,6,4,3] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => [1,6,2,4,3,5] => ? = 5 - 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => [1,4,3,6,2,5] => ? = 3 - 1
[1,2,6,3,5,4] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => [1,5,2,4,3,6] => ? = 4 - 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => [1,4,3,6,2,5] => ? = 3 - 1
[1,2,6,4,5,3] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => [1,5,2,4,3,6] => ? = 4 - 1
[1,2,6,5,3,4] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => [1,5,2,4,3,6] => ? = 4 - 1
[1,2,6,5,4,3] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => [1,5,2,4,3,6] => ? = 4 - 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,3,2,6,4,5] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => [1,5,2,6,3,4] => ? = 4 - 1
[1,3,2,6,5,4] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => [1,5,2,6,3,4] => ? = 4 - 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
[1,3,4,6,2,5] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => [1,5,2,6,3,4] => ? = 4 - 1
[1,3,4,6,5,2] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => [1,5,2,6,3,4] => ? = 4 - 1
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => [1,6,2,4,3,5] => ? = 5 - 1
[1,3,5,2,6,4] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => [1,4,3,5,2,6] => ? = 3 - 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => [1,6,2,4,3,5] => ? = 5 - 1
[1,3,5,4,6,2] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => [1,4,3,5,2,6] => ? = 3 - 1
[1,3,5,6,2,4] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => [1,6,2,4,3,5] => ? = 5 - 1
[1,3,5,6,4,2] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => [1,6,2,4,3,5] => ? = 5 - 1
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => [1,4,3,6,2,5] => ? = 3 - 1
[1,3,6,2,5,4] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => [1,5,2,4,3,6] => ? = 4 - 1
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => [1,4,3,6,2,5] => ? = 3 - 1
[1,3,6,4,5,2] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => [1,5,2,4,3,6] => ? = 4 - 1
[1,3,6,5,2,4] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => [1,5,2,4,3,6] => ? = 4 - 1
[1,3,6,5,4,2] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => [1,5,2,4,3,6] => ? = 4 - 1
[1,4,2,3,5,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => [1,6,2,5,3,4] => ? = 5 - 1
Description
The number of inversions of the second entry of a permutation. This is, for a permutation $\pi$ of length $n$, $$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001927Sparre Andersen's number of positives of a signed permutation.