- FindStat

Identifier

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words

Images

              [[1]] => [[1]] => [1] => [1] => 
            
              [[1,0],[0,1]] => [[1,1],[2]] => [3,1,2] => [3,1,2] => 00
            
              [[0,1],[1,0]] => [[1,2],[2]] => [2,1,3] => [3,2,1] => 00
            
              [[1,0,0],[0,1,0],[0,0,1]] => [[1,1,1],[2,2],[3]] => [6,4,5,1,2,3] => [5,6,1,3,4,2] => 00000
            
              [[0,1,0],[1,0,0],[0,0,1]] => [[1,1,2],[2,2],[3]] => [6,3,4,1,2,5] => [5,6,3,4,1,2] => 00000
            
              [[1,0,0],[0,0,1],[0,1,0]] => [[1,1,1],[2,3],[3]] => [5,4,6,1,2,3] => [5,6,1,3,2,4] => 00000
            
              [[0,1,0],[1,-1,1],[0,1,0]] => [[1,1,2],[2,3],[3]] => [5,3,6,1,2,4] => [5,6,3,1,2,4] => 00000
            
              [[0,0,1],[1,0,0],[0,1,0]] => [[1,1,3],[2,3],[3]] => [4,3,5,1,2,6] => [5,6,3,2,4,1] => 00000
            
              [[0,1,0],[0,0,1],[1,0,0]] => [[1,2,2],[2,3],[3]] => [5,2,6,1,3,4] => [5,3,6,1,2,4] => 00000
            
              [[0,0,1],[0,1,0],[1,0,0]] => [[1,2,3],[2,3],[3]] => [4,2,5,1,3,6] => [5,3,6,2,4,1] => 00000
            
              [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => [[1,1,1,1],[2,2,2],[3,3],[4]] => [10,8,9,5,6,7,1,2,3,4] => [8,9,10,1,5,6,7,3,4,2] => 000000000
            
              [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => [[1,1,1,2],[2,2,2],[3,3],[4]] => [10,8,9,4,5,6,1,2,3,7] => [8,9,10,5,6,7,1,3,4,2] => 000000000
            
              [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,1],[2,2,3],[3,3],[4]] => [10,7,8,5,6,9,1,2,3,4] => [8,9,10,1,5,6,3,4,7,2] => 000000000
            
              [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,2],[2,2,3],[3,3],[4]] => [10,7,8,4,5,9,1,2,3,6] => [8,9,10,5,6,1,3,4,7,2] => 000000000
            
              [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,3],[2,2,3],[3,3],[4]] => [10,6,7,4,5,8,1,2,3,9] => [8,9,10,5,6,3,4,7,1,2] => 000000000
            
              [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]] => [[1,1,2,2],[2,2,3],[3,3],[4]] => [10,7,8,3,4,9,1,2,5,6] => [8,9,5,6,10,1,3,4,7,2] => 000000000
            
              [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => [[1,1,2,3],[2,2,3],[3,3],[4]] => [10,6,7,3,4,8,1,2,5,9] => [8,9,5,6,10,3,4,7,1,2] => 000000000
            
              [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => [[1,1,1,1],[2,2,2],[3,4],[4]] => [9,8,10,5,6,7,1,2,3,4] => [8,9,10,1,5,6,7,3,2,4] => 000000000
            
              [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => [[1,1,1,2],[2,2,2],[3,4],[4]] => [9,8,10,4,5,6,1,2,3,7] => [8,9,10,5,6,7,1,3,2,4] => 000000000
            
              [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,1],[2,2,3],[3,4],[4]] => [9,7,10,5,6,8,1,2,3,4] => [8,9,10,1,5,6,3,7,2,4] => 000000000
            
              [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,2],[2,2,3],[3,4],[4]] => [9,7,10,4,5,8,1,2,3,6] => [8,9,10,5,6,1,3,7,2,4] => 000000000
            
              [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,3],[2,2,3],[3,4],[4]] => [9,6,10,4,5,7,1,2,3,8] => [8,9,10,5,6,3,7,1,2,4] => 000000000
            
              [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => [[1,1,2,2],[2,2,3],[3,4],[4]] => [9,7,10,3,4,8,1,2,5,6] => [8,9,5,6,10,1,3,7,2,4] => 000000000
            
              [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => [[1,1,2,3],[2,2,3],[3,4],[4]] => [9,6,10,3,4,7,1,2,5,8] => [8,9,5,6,10,3,7,1,2,4] => 000000000
            
              [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => [[1,1,1,1],[2,2,4],[3,4],[4]] => [8,7,9,5,6,10,1,2,3,4] => [8,9,10,1,5,6,3,2,4,7] => 000000000
            
              [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]] => [[1,1,1,2],[2,2,4],[3,4],[4]] => [8,7,9,4,5,10,1,2,3,6] => [8,9,10,5,6,1,3,2,4,7] => 000000000
            
              [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]] => [[1,1,1,3],[2,2,4],[3,4],[4]] => [8,6,9,4,5,10,1,2,3,7] => [8,9,10,5,6,3,1,2,4,7] => 000000000
            
              [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => [[1,1,1,4],[2,2,4],[3,4],[4]] => [7,6,8,4,5,9,1,2,3,10] => [8,9,10,5,6,3,2,4,7,1] => 000000000
            
              [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]] => [[1,1,2,2],[2,2,4],[3,4],[4]] => [8,7,9,3,4,10,1,2,5,6] => [8,9,5,6,10,1,3,2,4,7] => 000000000
            
              [[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]] => [[1,1,2,3],[2,2,4],[3,4],[4]] => [8,6,9,3,4,10,1,2,5,7] => [8,9,5,6,10,3,1,2,4,7] => 000000000
            
              [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => [[1,1,2,4],[2,2,4],[3,4],[4]] => [7,6,8,3,4,9,1,2,5,10] => [8,9,5,6,10,3,2,4,7,1] => 000000000
            
              [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => [[1,1,1,1],[2,3,3],[3,4],[4]] => [9,6,10,5,7,8,1,2,3,4] => [8,9,10,1,5,3,6,7,2,4] => 000000000
            
              [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]] => [[1,1,1,2],[2,3,3],[3,4],[4]] => [9,6,10,4,7,8,1,2,3,5] => [8,9,10,5,1,3,6,7,2,4] => 000000000
            
              [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]] => [[1,1,1,3],[2,3,3],[3,4],[4]] => [9,5,10,4,6,7,1,2,3,8] => [8,9,10,5,3,6,7,1,2,4] => 000000000
            
              [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]] => [[1,1,2,2],[2,3,3],[3,4],[4]] => [9,6,10,3,7,8,1,2,4,5] => [8,9,5,10,1,3,6,7,2,4] => 000000000
            
              [[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]] => [[1,1,2,3],[2,3,3],[3,4],[4]] => [9,5,10,3,6,7,1,2,4,8] => [8,9,5,10,3,6,7,1,2,4] => 000000000
            
              [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,1],[2,3,4],[3,4],[4]] => [8,6,9,5,7,10,1,2,3,4] => [8,9,10,1,5,3,6,2,4,7] => 000000000
            
              [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,2],[2,3,4],[3,4],[4]] => [8,6,9,4,7,10,1,2,3,5] => [8,9,10,5,1,3,6,2,4,7] => 000000000
            
              [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,3],[2,3,4],[3,4],[4]] => [8,5,9,4,6,10,1,2,3,7] => [8,9,10,5,3,6,1,2,4,7] => 000000000
            
              [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => [[1,1,1,4],[2,3,4],[3,4],[4]] => [7,5,8,4,6,9,1,2,3,10] => [8,9,10,5,3,6,2,4,7,1] => 000000000
            
              [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]] => [[1,1,2,2],[2,3,4],[3,4],[4]] => [8,6,9,3,7,10,1,2,4,5] => [8,9,5,10,1,3,6,2,4,7] => 000000000
            
              [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]] => [[1,1,2,3],[2,3,4],[3,4],[4]] => [8,5,9,3,6,10,1,2,4,7] => [8,9,5,10,3,6,1,2,4,7] => 000000000
            
              [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]] => [[1,1,2,4],[2,3,4],[3,4],[4]] => [7,5,8,3,6,9,1,2,4,10] => [8,9,5,10,3,6,2,4,7,1] => 000000000
            
              [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]] => [[1,1,3,3],[2,3,4],[3,4],[4]] => [8,4,9,3,5,10,1,2,6,7] => [8,9,5,3,6,10,1,2,4,7] => 000000000
            
              [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]] => [[1,1,3,4],[2,3,4],[3,4],[4]] => [7,4,8,3,5,9,1,2,6,10] => [8,9,5,3,6,10,2,4,7,1] => 000000000
            
              [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] => [[1,2,2,2],[2,3,3],[3,4],[4]] => [9,6,10,2,7,8,1,3,4,5] => [8,5,9,10,1,3,6,7,2,4] => 000000000
            
              [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]] => [[1,2,2,3],[2,3,3],[3,4],[4]] => [9,5,10,2,6,7,1,3,4,8] => [8,5,9,10,3,6,7,1,2,4] => 000000000
            
              [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]] => [[1,2,2,2],[2,3,4],[3,4],[4]] => [8,6,9,2,7,10,1,3,4,5] => [8,5,9,10,1,3,6,2,4,7] => 000000000
            
              [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => [[1,2,2,3],[2,3,4],[3,4],[4]] => [8,5,9,2,6,10,1,3,4,7] => [8,5,9,10,3,6,1,2,4,7] => 000000000
            
              [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => [[1,2,2,4],[2,3,4],[3,4],[4]] => [7,5,8,2,6,9,1,3,4,10] => [8,5,9,10,3,6,2,4,7,1] => 000000000
            
              [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]] => [[1,2,3,3],[2,3,4],[3,4],[4]] => [8,4,9,2,5,10,1,3,6,7] => [8,5,9,3,6,10,1,2,4,7] => 000000000
            
              [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => [[1,2,3,4],[2,3,4],[3,4],[4]] => [7,4,8,2,5,9,1,3,6,10] => [8,5,9,3,6,10,2,4,7,1] => 000000000

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.

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).

Map

Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$\pi^{-1}c$ where

$c = (1,\ldots,n)$ is the long cycle.

Map

connectivity set

Description

The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is

$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$
For

$n > 1$ it can also be described as the set of occurrences of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$
or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$
see [3].
The permutation is connected, when the connectivity set is empty.