- 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
St000174: 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 flush statistic of a semistandard tableau.
Let

$T$ be a tableaux with

$r$ rows such that each row is longer than the row beneath it by at least one box. Let

$1 \leq i < k \leq r+1$ and suppose

$l$ is the smallest integer greater than

$k$ such that there exists an

$l$ -segment in the

$(i+1)$ -st row of

$T$ . A

$k$ -segment in the

$i$ -th row of

$T$ is called flush if the leftmost box in the

$k$ -segment and the leftmost box of the

$l$ -segment are in the same column of

$T$ . If, however, no such

$l$ exists, then this

$k$ -segment is said to be flush if the number of boxes in the

$k$ -segment is equal to difference of the number of boxes between the

$i$ -th row and

$(i+1)$ -st row. The flush statistic is given by the number of

$k$ -segments in

$T$ .

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.