- FindStat

Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St000133

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [2,1] => 0
[2,3,1] => [2,1] => [2,1] => 0
[3,1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [2,1] => 0
[1,2,3,4] => [1,2,3] => [1,3,2] => 2
[1,2,4,3] => [1,2,3] => [1,3,2] => 2
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,3,2] => 2
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [2,3,1] => 0
[2,3,4,1] => [2,3,1] => [2,3,1] => 0
[2,4,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [2,3,1] => 0
[3,1,2,4] => [3,1,2] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => 0
[3,2,4,1] => [3,2,1] => [3,2,1] => 0
[3,4,1,2] => [3,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => 0
[4,1,2,3] => [1,2,3] => [1,3,2] => 2
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [2,3,1] => 0
[4,3,1,2] => [3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4] => [1,4,3,2] => 3
[1,2,3,5,4] => [1,2,3,4] => [1,4,3,2] => 3
[1,2,4,3,5] => [1,2,4,3] => [1,4,3,2] => 3
[1,2,4,5,3] => [1,2,4,3] => [1,4,3,2] => 3
[1,2,5,3,4] => [1,2,3,4] => [1,4,3,2] => 3
[1,2,5,4,3] => [1,2,4,3] => [1,4,3,2] => 3
[1,3,2,4,5] => [1,3,2,4] => [1,4,3,2] => 3
[1,3,2,5,4] => [1,3,2,4] => [1,4,3,2] => 3
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => 3
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => 3
[1,3,5,2,4] => [1,3,2,4] => [1,4,3,2] => 3
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => 3
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => 3
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => 3
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 3
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 3
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => 3
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => 3

Description

The "bounce" of a permutation.

Matching statistic: St000051

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [2,1] => [1] => [.,.]
 => 0
[2,1] => [1,2] => [1] => [.,.]
 => 0
[1,2,3] => [3,2,1] => [2,1] => [[.,.],.]
 => 1
[1,3,2] => [2,3,1] => [2,1] => [[.,.],.]
 => 1
[2,1,3] => [3,1,2] => [1,2] => [.,[.,.]]
 => 0
[2,3,1] => [1,3,2] => [1,2] => [.,[.,.]]
 => 0
[3,1,2] => [2,1,3] => [2,1] => [[.,.],.]
 => 1
[3,2,1] => [1,2,3] => [1,2] => [.,[.,.]]
 => 0
[1,2,3,4] => [4,3,2,1] => [3,2,1] => [[[.,.],.],.]
 => 2
[1,2,4,3] => [3,4,2,1] => [3,2,1] => [[[.,.],.],.]
 => 2
[1,3,2,4] => [4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => 2
[1,3,4,2] => [2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
 => 2
[1,4,2,3] => [3,2,4,1] => [3,2,1] => [[[.,.],.],.]
 => 2
[1,4,3,2] => [2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
 => 2
[2,1,3,4] => [4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => 1
[2,1,4,3] => [3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
 => 1
[2,3,1,4] => [4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 0
[2,3,4,1] => [1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 0
[2,4,1,3] => [3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
 => 1
[2,4,3,1] => [1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
 => 0
[3,1,2,4] => [4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 1
[3,1,4,2] => [2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 1
[3,2,1,4] => [4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
 => 0
[3,2,4,1] => [1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
 => 0
[3,4,1,2] => [2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
 => 1
[3,4,2,1] => [1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
 => 0
[4,1,2,3] => [3,2,1,4] => [3,2,1] => [[[.,.],.],.]
 => 2
[4,1,3,2] => [2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
 => 2
[4,2,1,3] => [3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
 => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
 => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
 => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
 => 0
[1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 3
[1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 3
[1,2,4,3,5] => [5,3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 3
[1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 3
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 3
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 3
[1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 3
[1,3,4,2,5] => [5,2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 3
[1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 3
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 3
[1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 3
[1,4,2,5,3] => [3,5,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 3
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3
[1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 3
[1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3

Description

The size of the left subtree of a binary tree.

Matching statistic: St001227

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1] => [1] => [1,0]
 => 0
[2,1] => [1] => [1] => [1,0]
 => 0
[1,2,3] => [1,2] => [1,2] => [1,0,1,0]
 => 1
[1,3,2] => [1,2] => [1,2] => [1,0,1,0]
 => 1
[2,1,3] => [2,1] => [2,1] => [1,1,0,0]
 => 0
[2,3,1] => [2,1] => [2,1] => [1,1,0,0]
 => 0
[3,1,2] => [1,2] => [1,2] => [1,0,1,0]
 => 1
[3,2,1] => [2,1] => [2,1] => [1,1,0,0]
 => 0
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[1,3,2,4] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[1,4,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[2,1,3,4] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[2,3,1,4] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[2,3,4,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[2,4,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[2,4,3,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[3,1,2,4] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[3,1,4,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[3,2,4,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[3,4,1,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[4,1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[4,2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[4,2,3,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[4,3,1,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3
[1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3

Description

The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.

Matching statistic: St000007

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1] => [1] => [1] => 1 = 0 + 1
[2,1] => [1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2] => [1,2] => [2,1] => 2 = 1 + 1
[1,3,2] => [1,2] => [1,2] => [2,1] => 2 = 1 + 1
[2,1,3] => [2,1] => [2,1] => [1,2] => 1 = 0 + 1
[2,3,1] => [2,1] => [2,1] => [1,2] => 1 = 0 + 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 2 = 1 + 1
[3,2,1] => [2,1] => [2,1] => [1,2] => 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[1,2,4,3] => [1,2,3] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[1,4,2,3] => [1,2,3] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [2,3,1] => [1,2,3] => 1 = 0 + 1
[2,3,4,1] => [2,3,1] => [2,3,1] => [1,2,3] => 1 = 0 + 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [2,3,1] => [1,2,3] => 1 = 0 + 1
[3,1,2,4] => [3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [2,1,3] => 1 = 0 + 1
[3,2,4,1] => [3,2,1] => [3,2,1] => [2,1,3] => 1 = 0 + 1
[3,4,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [2,1,3] => 1 = 0 + 1
[4,1,2,3] => [1,2,3] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 3 = 2 + 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [2,3,1] => [1,2,3] => 1 = 0 + 1
[4,3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [2,1,3] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000054

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [2,1] => [1] => [1] => 1 = 0 + 1
[2,1] => [1,2] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [3,2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,3,2] => [2,3,1] => [2,1] => [2,1] => 2 = 1 + 1
[2,1,3] => [3,1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,2] => [1,2] => 1 = 0 + 1
[3,1,2] => [2,1,3] => [2,1] => [2,1] => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2] => [1,2] => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,2,4,3] => [3,4,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,3,2,4] => [4,2,3,1] => [2,3,1] => [3,1,2] => 3 = 2 + 1
[1,3,4,2] => [2,4,3,1] => [2,3,1] => [3,1,2] => 3 = 2 + 1
[1,4,2,3] => [3,2,4,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,4,3,2] => [2,3,4,1] => [2,3,1] => [3,1,2] => 3 = 2 + 1
[2,1,3,4] => [4,3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[2,1,4,3] => [3,4,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[2,3,1,4] => [4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 0 + 1
[2,3,4,1] => [1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 0 + 1
[2,4,1,3] => [3,1,4,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[2,4,3,1] => [1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 0 + 1
[3,1,2,4] => [4,2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[3,2,1,4] => [4,1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,4,1,2] => [2,1,4,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[4,1,2,3] => [3,2,1,4] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[4,1,3,2] => [2,3,1,4] => [2,3,1] => [3,1,2] => 3 = 2 + 1
[4,2,1,3] => [3,1,2,4] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 0 + 1
[4,3,1,2] => [2,1,3,4] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 4 = 3 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => [4,3,1,2] => 4 = 3 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => [4,3,1,2] => 4 = 3 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 4 = 3 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => [4,2,3,1] => 4 = 3 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [2,4,3,1] => [4,1,3,2] => 4 = 3 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => [4,1,3,2] => 4 = 3 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,3,1] => [4,2,3,1] => 4 = 3 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => [4,1,3,2] => 4 = 3 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [3,2,4,1] => [4,2,1,3] => 4 = 3 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [3,2,4,1] => [4,2,1,3] => 4 = 3 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 4 = 3 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => [4,1,2,3] => 4 = 3 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,4,1] => [4,2,1,3] => 4 = 3 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => [4,1,2,3] => 4 = 3 + 1

Description

The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation

$\pi$ of

$n$ , together with its rotations, obtained by conjugating with the long cycle

$(1,\dots,n)$ . Drawing the labels

$1$ to

$n$ in this order on a circle, and the arcs

$(i, \pi(i))$ as straight lines, the rotation of

$\pi$ is obtained by replacing each number

$i$ by

$(i\bmod n) +1$ . Then,

$\pi(1)-1$ is the number of rotations of

$\pi$ where the arc

$(1, \pi(1))$ is a deficiency. In particular, if

$O(\pi)$ is the orbit of rotations of

$\pi$ , then the number of deficiencies of

$\pi$ equals

$\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).$

Matching statistic: St000740

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [2,1] => [1] => [1] => 1 = 0 + 1
[2,1] => [1,2] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [3,2,1] => [2,1] => [1,2] => 2 = 1 + 1
[1,3,2] => [2,3,1] => [2,1] => [1,2] => 2 = 1 + 1
[2,1,3] => [3,1,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,2] => [2,1] => 1 = 0 + 1
[3,1,2] => [2,1,3] => [2,1] => [1,2] => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2] => [2,1] => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[1,2,4,3] => [3,4,2,1] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[1,3,2,4] => [4,2,3,1] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[1,3,4,2] => [2,4,3,1] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[1,4,2,3] => [3,2,4,1] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[1,4,3,2] => [2,3,4,1] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[2,1,3,4] => [4,3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[2,1,4,3] => [3,4,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[2,3,1,4] => [4,1,3,2] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[2,3,4,1] => [1,4,3,2] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[2,4,1,3] => [3,1,4,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[2,4,3,1] => [1,3,4,2] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[3,1,2,4] => [4,2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[3,2,1,4] => [4,1,2,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[3,4,1,2] => [2,1,4,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[4,1,2,3] => [3,2,1,4] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[4,1,3,2] => [2,3,1,4] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[4,2,1,3] => [3,1,2,4] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[4,3,1,2] => [2,1,3,4] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => [3,2,1,4] => 4 = 3 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => [3,2,1,4] => 4 = 3 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [3,4,2,1] => [3,1,2,4] => 4 = 3 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => [3,1,2,4] => 4 = 3 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,2,1] => [3,2,1,4] => 4 = 3 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => [3,1,2,4] => 4 = 3 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [4,2,3,1] => [2,3,1,4] => 4 = 3 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => [2,3,1,4] => 4 = 3 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [2,4,3,1] => [1,3,2,4] => 4 = 3 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => [1,3,2,4] => 4 = 3 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,3,1] => [2,3,1,4] => 4 = 3 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => [1,3,2,4] => 4 = 3 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [3,2,4,1] => [2,1,3,4] => 4 = 3 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [3,2,4,1] => [2,1,3,4] => 4 = 3 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,1] => [1,2,3,4] => 4 = 3 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => [1,2,3,4] => 4 = 3 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,4,1] => [2,1,3,4] => 4 = 3 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => [1,2,3,4] => 4 = 3 + 1

Description

The last entry of a permutation. This statistic is undefined for the empty permutation.

Matching statistic: St001184

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1] => [1] => [1,0]
 => 1 = 0 + 1
[2,1] => [1] => [1] => [1,0]
 => 1 = 0 + 1
[1,2,3] => [1,2] => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[1,3,2] => [1,2] => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[2,1,3] => [2,1] => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[2,3,1] => [2,1] => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[3,1,2] => [1,2] => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[3,2,1] => [2,1] => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,4,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,2,4] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,4,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,3,4] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[2,3,4,1] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[2,4,1,3] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[3,1,2,4] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[3,2,4,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[3,4,1,2] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[4,1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[4,1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[4,2,1,3] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[4,3,1,2] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,4,5,3] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,5,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,5,4,3] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,4,2,5] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,4,5,2] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,5,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,5,4,2] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,2,3,5] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,2,5,3] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,3,5,2] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,5,2,3] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,5,3,2] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1

Description

Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.

Matching statistic: St000066

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%

Values

[1,2] => [2,1] => [1] => [[1]]
 => 1 = 0 + 1
[2,1] => [1,2] => [1] => [[1]]
 => 1 = 0 + 1
[1,2,3] => [3,2,1] => [2,1] => [[0,1],[1,0]]
 => 2 = 1 + 1
[1,3,2] => [2,3,1] => [2,1] => [[0,1],[1,0]]
 => 2 = 1 + 1
[2,1,3] => [3,1,2] => [1,2] => [[1,0],[0,1]]
 => 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,2] => [[1,0],[0,1]]
 => 1 = 0 + 1
[3,1,2] => [2,1,3] => [2,1] => [[0,1],[1,0]]
 => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2] => [[1,0],[0,1]]
 => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 3 = 2 + 1
[1,2,4,3] => [3,4,2,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 3 = 2 + 1
[1,3,2,4] => [4,2,3,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 3 = 2 + 1
[1,3,4,2] => [2,4,3,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 3 = 2 + 1
[1,4,2,3] => [3,2,4,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 3 = 2 + 1
[1,4,3,2] => [2,3,4,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 3 = 2 + 1
[2,1,3,4] => [4,3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 2 = 1 + 1
[2,1,4,3] => [3,4,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 2 = 1 + 1
[2,3,1,4] => [4,1,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 1 = 0 + 1
[2,3,4,1] => [1,4,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 1 = 0 + 1
[2,4,1,3] => [3,1,4,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 2 = 1 + 1
[2,4,3,1] => [1,3,4,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 1 = 0 + 1
[3,1,2,4] => [4,2,1,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 1 + 1
[3,2,1,4] => [4,1,2,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 1 = 0 + 1
[3,4,1,2] => [2,1,4,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 1 = 0 + 1
[4,1,2,3] => [3,2,1,4] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 3 = 2 + 1
[4,1,3,2] => [2,3,1,4] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 3 = 2 + 1
[4,2,1,3] => [3,1,2,4] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 1 = 0 + 1
[4,3,1,2] => [2,1,3,4] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 4 = 3 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 4 = 3 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 4 = 3 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 4 = 3 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 4 = 3 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 4 = 3 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 4 = 3 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 4 = 3 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 4 = 3 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 4 = 3 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 4 = 3 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 4 = 3 + 1
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [5,3,6,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [5,3,6,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [5,3,6,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [5,3,6,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [4,6,3,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => [4,6,3,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => [4,6,3,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [3,6,4,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [3,6,4,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => [3,6,4,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => [4,6,3,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,2,5,7,4,6,3] => [3,6,4,7,5,2,1] => [3,6,4,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,2,7,4,6,3,5] => [5,3,6,4,7,2,1] => [5,3,6,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,2,7,5,3,6,4] => [4,6,3,5,7,2,1] => [4,6,3,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,2,7,5,4,6,3] => [3,6,4,5,7,2,1] => [3,6,4,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,3,4,6,2,5,7] => [7,5,2,6,4,3,1] => [5,2,6,4,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,3,4,6,2,7,5] => [5,7,2,6,4,3,1] => [5,2,6,4,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,3,4,6,7,2,5] => [5,2,7,6,4,3,1] => [5,2,6,4,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,3,4,7,6,2,5] => [5,2,6,7,4,3,1] => [5,2,6,4,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,3,5,2,4,6,7] => [7,6,4,2,5,3,1] => [6,4,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,3,5,2,4,7,6] => [6,7,4,2,5,3,1] => [6,4,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,3,5,2,6,4,7] => [7,4,6,2,5,3,1] => [4,6,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,3,5,2,6,7,4] => [4,7,6,2,5,3,1] => [4,6,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,3,5,2,7,4,6] => [6,4,7,2,5,3,1] => [6,4,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,3,5,2,7,6,4] => [4,6,7,2,5,3,1] => [4,6,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,3,5,6,2,4,7] => [7,4,2,6,5,3,1] => [4,2,6,5,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,3,5,6,2,7,4] => [4,7,2,6,5,3,1] => [4,2,6,5,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,3,5,6,7,2,4] => [4,2,7,6,5,3,1] => [4,2,6,5,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,3,5,7,2,4,6] => [6,4,2,7,5,3,1] => [6,4,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,3,5,7,2,6,4] => [4,6,2,7,5,3,1] => [4,6,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1
[1,3,5,7,6,2,4] => [4,2,6,7,5,3,1] => [4,2,6,5,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,3,6,2,4,5,7] => [7,5,4,2,6,3,1] => [5,4,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,2,4,7,5] => [5,7,4,2,6,3,1] => [5,4,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,2,5,4,7] => [7,4,5,2,6,3,1] => [4,5,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,2,5,7,4] => [4,7,5,2,6,3,1] => [4,5,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,2,7,4,5] => [5,4,7,2,6,3,1] => [5,4,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,2,7,5,4] => [4,5,7,2,6,3,1] => [4,5,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,4,2,5,7] => [7,5,2,4,6,3,1] => [5,2,4,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,4,2,7,5] => [5,7,2,4,6,3,1] => [5,2,4,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,4,7,2,5] => [5,2,7,4,6,3,1] => [5,2,4,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,5,2,4,7] => [7,4,2,5,6,3,1] => [4,2,5,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,5,2,7,4] => [4,7,2,5,6,3,1] => [4,2,5,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,5,7,2,4] => [4,2,7,5,6,3,1] => [4,2,5,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,7,2,4,5] => [5,4,2,7,6,3,1] => [5,4,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,7,2,5,4] => [4,5,2,7,6,3,1] => [4,5,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,7,4,2,5] => [5,2,4,7,6,3,1] => [5,2,4,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,6,7,5,2,4] => [4,2,5,7,6,3,1] => [4,2,5,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 5 + 1
[1,3,7,4,6,2,5] => [5,2,6,4,7,3,1] => [5,2,6,4,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 5 + 1
[1,3,7,5,2,4,6] => [6,4,2,5,7,3,1] => [6,4,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,3,7,5,2,6,4] => [4,6,2,5,7,3,1] => [4,6,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 5 + 1

Description

The column of the unique '1' in the first row of the alternating sign matrix. The generating function of this statistic is given by

$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$ see [2].

Matching statistic: St000193

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00003: Alternating sign matrices —rotate counterclockwise⟶ Alternating sign matrices
St000193: Alternating sign matrices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 83%

Values

[1,2] => [1] => [[1]]
 => [[1]]
 => 1 = 0 + 1
[2,1] => [1] => [[1]]
 => [[1]]
 => 1 = 0 + 1
[1,2,3] => [1,2] => [[1,0],[0,1]]
 => [[0,1],[1,0]]
 => 2 = 1 + 1
[1,3,2] => [1,2] => [[1,0],[0,1]]
 => [[0,1],[1,0]]
 => 2 = 1 + 1
[2,1,3] => [2,1] => [[0,1],[1,0]]
 => [[1,0],[0,1]]
 => 1 = 0 + 1
[2,3,1] => [2,1] => [[0,1],[1,0]]
 => [[1,0],[0,1]]
 => 1 = 0 + 1
[3,1,2] => [1,2] => [[1,0],[0,1]]
 => [[0,1],[1,0]]
 => 2 = 1 + 1
[3,2,1] => [2,1] => [[0,1],[1,0]]
 => [[1,0],[0,1]]
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 3 = 2 + 1
[1,2,4,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 3 = 2 + 1
[1,3,2,4] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => [[0,1,0],[0,0,1],[1,0,0]]
 => 3 = 2 + 1
[1,3,4,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => [[0,1,0],[0,0,1],[1,0,0]]
 => 3 = 2 + 1
[1,4,2,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 3 = 2 + 1
[1,4,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => [[0,1,0],[0,0,1],[1,0,0]]
 => 3 = 2 + 1
[2,1,3,4] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 1 = 0 + 1
[2,3,4,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 1 = 0 + 1
[2,4,1,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 1 = 0 + 1
[3,1,2,4] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 1 = 0 + 1
[3,2,4,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 1 = 0 + 1
[3,4,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 1 = 0 + 1
[4,1,2,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 3 = 2 + 1
[4,1,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => [[0,1,0],[0,0,1],[1,0,0]]
 => 3 = 2 + 1
[4,2,1,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 1 = 0 + 1
[4,3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,2,4,5,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,2,5,3,4] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,2,5,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,3,4,2,5] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,3,4,5,2] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,3,5,2,4] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,3,5,4,2] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => 4 = 3 + 1
[1,4,2,3,5] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 4 = 3 + 1
[1,4,2,5,3] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 4 = 3 + 1
[1,4,3,5,2] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 4 = 3 + 1
[1,4,5,2,3] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 4 = 3 + 1
[1,4,5,3,2] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 4 = 3 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,5,6,7,4] => [1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,5,7,4,6] => [1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,5,7,6,4] => [1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,6,4,7,5] => [1,2,3,6,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,6,5,4,7] => [1,2,3,6,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[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]]
 => ? = 5 + 1
[1,2,3,6,5,7,4] => [1,2,3,6,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[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]]
 => ? = 5 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,6,7,5,4] => [1,2,3,6,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[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]]
 => ? = 5 + 1
[1,2,3,7,4,5,6] => [1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,7,4,6,5] => [1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,7,5,4,6] => [1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,7,5,6,4] => [1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,7,6,4,5] => [1,2,3,6,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[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]]
 => ? = 5 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6] => [[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],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,6] => [[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],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,5,6,7,3] => [1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,5,7,3,6] => [1,2,4,5,3,6] => [[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],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,5,7,6,3] => [1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,6,3,5,7] => [1,2,4,6,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => [[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]]
 => ? = 5 + 1
[1,2,4,6,3,7,5] => [1,2,4,6,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => [[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]]
 => ? = 5 + 1
[1,2,4,6,5,3,7] => [1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,6,5,7,3] => [1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,6,7,3,5] => [1,2,4,6,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => [[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]]
 => ? = 5 + 1
[1,2,4,6,7,5,3] => [1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,7,3,5,6] => [1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,7,3,6,5] => [1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,7,5,3,6] => [1,2,4,5,3,6] => [[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],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,7,5,6,3] => [1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,4,7,6,3,5] => [1,2,4,6,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => [[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]]
 => ? = 5 + 1
[1,2,4,7,6,5,3] => [1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6] => [[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],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,6] => [[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],[0,0,0,0,0,1]]
 => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 5 + 1

Description

The row of the unique '1' in the first column of the alternating sign matrix.

Matching statistic: St001557

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%

Values

[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,3,2] => [2,3,1] => [3,1,2] => [2,3,1] => 1
[2,1,3] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [2,1,3] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,2,4,3] => [3,4,2,1] => [4,1,3,2] => [2,4,3,1] => 2
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 2
[1,3,4,2] => [2,4,3,1] => [3,1,4,2] => [2,4,1,3] => 2
[1,4,2,3] => [3,2,4,1] => [4,3,1,2] => [3,4,2,1] => 2
[1,4,3,2] => [2,3,4,1] => [3,4,1,2] => [3,4,1,2] => 2
[2,1,3,4] => [4,3,1,2] => [4,2,3,1] => [4,2,3,1] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [4,1,3,2] => 0
[2,3,4,1] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => [3,2,4,1] => 1
[2,4,3,1] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 0
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [4,2,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [3,1,2,4] => [2,3,1,4] => 1
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [2,1,3,4] => [2,1,3,4] => 0
[3,4,1,2] => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 1
[3,4,2,1] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 0
[4,1,2,3] => [3,2,1,4] => [1,4,3,2] => [1,4,3,2] => 2
[4,1,3,2] => [2,3,1,4] => [1,3,4,2] => [1,4,2,3] => 2
[4,2,1,3] => [3,1,2,4] => [1,4,2,3] => [1,3,4,2] => 1
[4,2,3,1] => [1,3,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,1,2] => [2,1,3,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 3
[1,2,3,5,4] => [4,5,3,2,1] => [5,1,4,3,2] => [2,5,4,3,1] => 3
[1,2,4,3,5] => [5,3,4,2,1] => [4,5,3,2,1] => [5,4,3,1,2] => 3
[1,2,4,5,3] => [3,5,4,2,1] => [4,1,5,3,2] => [2,5,4,1,3] => 3
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,1,3,2] => [3,5,4,2,1] => 3
[1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,3,2] => [3,5,4,1,2] => 3
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => [5,4,2,3,1] => 3
[1,3,2,5,4] => [4,5,2,3,1] => [5,1,3,4,2] => [2,5,3,4,1] => 3
[1,3,4,2,5] => [5,2,4,3,1] => [3,5,4,2,1] => [5,4,1,3,2] => 3
[1,3,4,5,2] => [2,5,4,3,1] => [3,1,5,4,2] => [2,5,1,4,3] => 3
[1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,4,2] => [3,5,2,4,1] => 3
[1,3,5,4,2] => [2,4,5,3,1] => [3,5,1,4,2] => [3,5,1,4,2] => 3
[1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => [5,4,2,1,3] => 3
[1,4,2,5,3] => [3,5,2,4,1] => [4,1,3,5,2] => [2,5,3,1,4] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [3,4,5,2,1] => [5,4,1,2,3] => 3
[1,4,3,5,2] => [2,5,3,4,1] => [3,1,4,5,2] => [2,5,1,3,4] => 3
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,1,5,2] => [3,5,2,1,4] => 3
[1,4,5,3,2] => [2,3,5,4,1] => [3,4,1,5,2] => [3,5,1,2,4] => 3
[1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 4
[1,2,3,4,6,5] => [5,6,4,3,2,1] => [6,1,5,4,3,2] => [2,6,5,4,3,1] => ? = 4
[1,2,3,5,4,6] => [6,4,5,3,2,1] => [5,6,4,3,2,1] => [6,5,4,3,1,2] => ? = 4
[1,2,3,5,6,4] => [4,6,5,3,2,1] => [5,1,6,4,3,2] => [2,6,5,4,1,3] => ? = 4
[1,2,3,6,4,5] => [5,4,6,3,2,1] => [6,5,1,4,3,2] => [3,6,5,4,2,1] => ? = 4
[1,2,3,6,5,4] => [4,5,6,3,2,1] => [5,6,1,4,3,2] => [3,6,5,4,1,2] => ? = 4
[1,2,4,3,5,6] => [6,5,3,4,2,1] => [6,4,5,3,2,1] => [6,5,4,2,3,1] => ? = 4
[1,2,4,3,6,5] => [5,6,3,4,2,1] => [6,1,4,5,3,2] => [2,6,5,3,4,1] => ? = 4
[1,2,4,5,3,6] => [6,3,5,4,2,1] => [4,6,5,3,2,1] => [6,5,4,1,3,2] => ? = 4
[1,2,4,5,6,3] => [3,6,5,4,2,1] => [4,1,6,5,3,2] => [2,6,5,1,4,3] => ? = 4
[1,2,4,6,3,5] => [5,3,6,4,2,1] => [6,4,1,5,3,2] => [3,6,5,2,4,1] => ? = 4
[1,2,4,6,5,3] => [3,5,6,4,2,1] => [4,6,1,5,3,2] => [3,6,5,1,4,2] => ? = 4
[1,2,5,3,4,6] => [6,4,3,5,2,1] => [5,4,6,3,2,1] => [6,5,4,2,1,3] => ? = 4
[1,2,5,3,6,4] => [4,6,3,5,2,1] => [5,1,4,6,3,2] => [2,6,5,3,1,4] => ? = 4
[1,2,5,4,3,6] => [6,3,4,5,2,1] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => ? = 4
[1,2,5,4,6,3] => [3,6,4,5,2,1] => [4,1,5,6,3,2] => [2,6,5,1,3,4] => ? = 4
[1,2,5,6,3,4] => [4,3,6,5,2,1] => [5,4,1,6,3,2] => [3,6,5,2,1,4] => ? = 4
[1,2,5,6,4,3] => [3,4,6,5,2,1] => [4,5,1,6,3,2] => [3,6,5,1,2,4] => ? = 4
[1,2,6,3,4,5] => [5,4,3,6,2,1] => [6,5,4,1,3,2] => [4,6,5,3,2,1] => ? = 4
[1,2,6,3,5,4] => [4,5,3,6,2,1] => [5,6,4,1,3,2] => [4,6,5,3,1,2] => ? = 4
[1,2,6,4,3,5] => [5,3,4,6,2,1] => [6,4,5,1,3,2] => [4,6,5,2,3,1] => ? = 4
[1,2,6,4,5,3] => [3,5,4,6,2,1] => [4,6,5,1,3,2] => [4,6,5,1,3,2] => ? = 4
[1,2,6,5,3,4] => [4,3,5,6,2,1] => [5,4,6,1,3,2] => [4,6,5,2,1,3] => ? = 4
[1,2,6,5,4,3] => [3,4,5,6,2,1] => [4,5,6,1,3,2] => [4,6,5,1,2,3] => ? = 4
[1,3,2,4,5,6] => [6,5,4,2,3,1] => [6,5,3,4,2,1] => [6,5,3,4,2,1] => ? = 4
[1,3,2,4,6,5] => [5,6,4,2,3,1] => [6,1,5,3,4,2] => [2,6,4,5,3,1] => ? = 4
[1,3,2,5,4,6] => [6,4,5,2,3,1] => [5,6,3,4,2,1] => [6,5,3,4,1,2] => ? = 4
[1,3,2,5,6,4] => [4,6,5,2,3,1] => [5,1,6,3,4,2] => [2,6,4,5,1,3] => ? = 4
[1,3,2,6,4,5] => [5,4,6,2,3,1] => [6,5,1,3,4,2] => [3,6,4,5,2,1] => ? = 4
[1,3,2,6,5,4] => [4,5,6,2,3,1] => [5,6,1,3,4,2] => [3,6,4,5,1,2] => ? = 4
[1,3,4,2,5,6] => [6,5,2,4,3,1] => [6,3,5,4,2,1] => [6,5,2,4,3,1] => ? = 4
[1,3,4,2,6,5] => [5,6,2,4,3,1] => [6,1,3,5,4,2] => [2,6,3,5,4,1] => ? = 4
[1,3,4,5,2,6] => [6,2,5,4,3,1] => [3,6,5,4,2,1] => [6,5,1,4,3,2] => ? = 4
[1,3,4,5,6,2] => [2,6,5,4,3,1] => [3,1,6,5,4,2] => [2,6,1,5,4,3] => ? = 4
[1,3,4,6,2,5] => [5,2,6,4,3,1] => [6,3,1,5,4,2] => [3,6,2,5,4,1] => ? = 4
[1,3,4,6,5,2] => [2,5,6,4,3,1] => [3,6,1,5,4,2] => [3,6,1,5,4,2] => ? = 4
[1,3,5,2,4,6] => [6,4,2,5,3,1] => [5,3,6,4,2,1] => [6,5,2,4,1,3] => ? = 4
[1,3,5,2,6,4] => [4,6,2,5,3,1] => [5,1,3,6,4,2] => [2,6,3,5,1,4] => ? = 4
[1,3,5,4,2,6] => [6,2,4,5,3,1] => [3,5,6,4,2,1] => [6,5,1,4,2,3] => ? = 4
[1,3,5,4,6,2] => [2,6,4,5,3,1] => [3,1,5,6,4,2] => [2,6,1,5,3,4] => ? = 4
[1,3,5,6,2,4] => [4,2,6,5,3,1] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => ? = 4
[1,3,5,6,4,2] => [2,4,6,5,3,1] => [3,5,1,6,4,2] => [3,6,1,5,2,4] => ? = 4
[1,3,6,2,4,5] => [5,4,2,6,3,1] => [6,5,3,1,4,2] => [4,6,3,5,2,1] => ? = 4
[1,3,6,2,5,4] => [4,5,2,6,3,1] => [5,6,3,1,4,2] => [4,6,3,5,1,2] => ? = 4
[1,3,6,4,2,5] => [5,2,4,6,3,1] => [6,3,5,1,4,2] => [4,6,2,5,3,1] => ? = 4
[1,3,6,4,5,2] => [2,5,4,6,3,1] => [3,6,5,1,4,2] => [4,6,1,5,3,2] => ? = 4
[1,3,6,5,2,4] => [4,2,5,6,3,1] => [5,3,6,1,4,2] => [4,6,2,5,1,3] => ? = 4
[1,3,6,5,4,2] => [2,4,5,6,3,1] => [3,5,6,1,4,2] => [4,6,1,5,2,3] => ? = 4
[1,4,2,3,5,6] => [6,5,3,2,4,1] => [6,4,3,5,2,1] => [6,5,3,2,4,1] => ? = 4
[1,4,2,3,6,5] => [5,6,3,2,4,1] => [6,1,4,3,5,2] => [2,6,4,3,5,1] => ? = 4

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 1 statistic also match your data. Click on any of them to see the details.

St001876The number of 2-regular simple modules in the incidence algebra of the lattice.