- FindStat

Identifier

Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000001: Permutations ⟶ ℤ

Values

[[1]] => [1] => 1

[[1,0],[0,1]] => [1,2] => 1

[[0,1],[1,0]] => [2,1] => 1

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

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

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

[[0,1,0],[1,-1,1],[0,1,0]] => [1,3,2] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => [4,3,2,1] => 16

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [4,3,2,1,5] => 16

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

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

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

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

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

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

[[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]] => [3,2,1,5,4] => 8

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Description

The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.

Map

to left key permutation

Description

Return the permutation of the left key of an alternating sign matrix.
This was originally defined by Lascoux and then further studied by Aval [1].