Identifier
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Images
[[1]] => [[1]] => [1] => [1] => [[1]]
[[1,0],[0,1]] => [[1,1],[2]] => [3,1,2] => [2,1,3] => [[1,3],[2]]
[[0,1],[1,0]] => [[1,2],[2]] => [2,1,3] => [3,1,2] => [[1,3],[2]]
[[1,0,0],[0,1,0],[0,0,1]] => [[1,1,1],[2,2],[3]] => [6,4,5,1,2,3] => [3,2,1,5,4,6] => [[1,4,6],[2,5],[3]]
[[0,1,0],[1,0,0],[0,0,1]] => [[1,1,2],[2,2],[3]] => [6,3,4,1,2,5] => [5,2,1,4,3,6] => [[1,4,6],[2,5],[3]]
[[1,0,0],[0,0,1],[0,1,0]] => [[1,1,1],[2,3],[3]] => [5,4,6,1,2,3] => [3,2,1,6,4,5] => [[1,4,6],[2,5],[3]]
[[0,1,0],[1,-1,1],[0,1,0]] => [[1,1,2],[2,3],[3]] => [5,3,6,1,2,4] => [4,2,1,6,3,5] => [[1,4,6],[2,5],[3]]
[[0,0,1],[1,0,0],[0,1,0]] => [[1,1,3],[2,3],[3]] => [4,3,5,1,2,6] => [6,2,1,5,3,4] => [[1,4,6],[2,5],[3]]
[[0,1,0],[0,0,1],[1,0,0]] => [[1,2,2],[2,3],[3]] => [5,2,6,1,3,4] => [4,3,1,6,2,5] => [[1,4,6],[2,5],[3]]
[[0,0,1],[0,1,0],[1,0,0]] => [[1,2,3],[2,3],[3]] => [4,2,5,1,3,6] => [6,3,1,5,2,4] => [[1,4,6],[2,5],[3]]
[[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] => [4,3,2,1,7,6,5,9,8,10] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [7,3,2,1,6,5,4,9,8,10] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [4,3,2,1,9,6,5,8,7,10] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [6,3,2,1,9,5,4,8,7,10] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [9,3,2,1,8,5,4,7,6,10] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [6,5,2,1,9,4,3,8,7,10] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [9,5,2,1,8,4,3,7,6,10] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [4,3,2,1,7,6,5,10,8,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [7,3,2,1,6,5,4,10,8,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [4,3,2,1,8,6,5,10,7,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [6,3,2,1,8,5,4,10,7,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [8,3,2,1,7,5,4,10,6,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [6,5,2,1,8,4,3,10,7,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [8,5,2,1,7,4,3,10,6,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [4,3,2,1,10,6,5,9,7,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [6,3,2,1,10,5,4,9,7,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [7,3,2,1,10,5,4,9,6,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [10,3,2,1,9,5,4,8,6,7] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [6,5,2,1,10,4,3,9,7,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [7,5,2,1,10,4,3,9,6,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [10,5,2,1,9,4,3,8,6,7] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [4,3,2,1,8,7,5,10,6,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [5,3,2,1,8,7,4,10,6,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [8,3,2,1,7,6,4,10,5,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [5,4,2,1,8,7,3,10,6,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [8,4,2,1,7,6,3,10,5,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [4,3,2,1,10,7,5,9,6,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [5,3,2,1,10,7,4,9,6,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [7,3,2,1,10,6,4,9,5,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [10,3,2,1,9,6,4,8,5,7] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [5,4,2,1,10,7,3,9,6,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [7,4,2,1,10,6,3,9,5,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [10,4,2,1,9,6,3,8,5,7] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [7,6,2,1,10,5,3,9,4,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [10,6,2,1,9,5,3,8,4,7] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [5,4,3,1,8,7,2,10,6,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [8,4,3,1,7,6,2,10,5,9] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [5,4,3,1,10,7,2,9,6,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [7,4,3,1,10,6,2,9,5,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [10,4,3,1,9,6,2,8,5,7] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [7,6,3,1,10,5,2,9,4,8] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[[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] => [10,6,3,1,9,5,2,8,4,7] => [[1,5,8,10],[2,6,9],[3,7],[4]]
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
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
Robinson-Schensted recording tableau
Description
Sends a permutation to its Robinson-Schensted recording tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding recording tableau.