- FindStat

Identifier

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000173: Semistandard tableaux ⟶ ℤ

Values

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

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$F_{1} = 2$

$F_{2} = 4 + 4\ q$

$F_{3} = 8 + 16\ q + 16\ q^{2} + 8\ q^{3}$

Description

The segment statistic of a semistandard tableau.
Let T be a tableau. A k-segment of T (in the ith row) is defined to be a maximal consecutive sequence of k-boxes in the ith row. Note that the possible i-boxes in the ith row are not considered to be i-segments. Then seg(T) is the total number of k-segments in T as k varies over all possible values.

Map

to alternating sign matrix

Description

Maps a permutation to its permutation matrix as an alternating sign matrix.

Map

permutation

Description

The permutation obtained by forgetting the colours.

Map

to semistandard tableau via monotone triangles

Description

The semistandard tableau corresponding the monotone triangle of an alternating sign matrix.
This is obtained by interpreting each row of the monotone triangle as an integer partition, and filling the cells of the smallest partition with ones, the second smallest with twos, and so on.