- FindStat

Identifier

Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000882: 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] => 1

[[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] => 1

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

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

[[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] => 1

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

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

[[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] => 1

[[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] => 1

[[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] => 2

[[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] => 2

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

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

[[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] => 2

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

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

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

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

[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => [4,3,2,1] => 8

[[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] => 1

[[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] => 1

[[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] => 1

[[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] => 1

[[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] => 1

[[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] => 1

[[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] => 2

[[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] => 1

[[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] => 1

[[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] => 2

[[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] => 2

[[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] => 1

[[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] => 3

[[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] => 2

[[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] => 2

[[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] => 2

[[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] => 3

[[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] => 3

[[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] => 8

[[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] => 1

[[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] => 1

[[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] => 1

[[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] => 1

[[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] => 1

[[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] => 2

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Description

The number of connected components of short braid edges in the graph of braid moves of a permutation.
Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form $s_i s_j \leftrightarrow s_j s_i$ for $|i-j| > 1$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2$ and $s_3s_1s_2$ for
$$(1243) = (12)(34)(23) = (34)(12)(23)$$
share an edge because they are obtained from each other by interchanging $s_1s_3 \leftrightarrow s_3s_1$.
This statistic counts the number connected components of such short braid moves among all reduced words.

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].