Identifier
Mp00001: to semistandard tableau via monotone trianglesSemistandard tableaux
Mp00070: Permutations Robinson-Schensted recording tableau
Images
=>
Cc0017;cc-rep-0Cc0019;cc-rep-1Cc0007;cc-rep-3
[[1]]=>[[1]]=>[1]=>[[1]] [[1,0],[0,1]]=>[[1,1],[2]]=>[3,1,2]=>[[1,3],[2]] [[0,1],[1,0]]=>[[1,2],[2]]=>[2,1,3]=>[[1,3],[2]] [[1,0,0],[0,1,0],[0,0,1]]=>[[1,1,1],[2,2],[3]]=>[6,4,5,1,2,3]=>[[1,3,6],[2,5],[4]] [[0,1,0],[1,0,0],[0,0,1]]=>[[1,1,2],[2,2],[3]]=>[6,3,4,1,2,5]=>[[1,3,6],[2,5],[4]] [[1,0,0],[0,0,1],[0,1,0]]=>[[1,1,1],[2,3],[3]]=>[5,4,6,1,2,3]=>[[1,3,6],[2,5],[4]] [[0,1,0],[1,-1,1],[0,1,0]]=>[[1,1,2],[2,3],[3]]=>[5,3,6,1,2,4]=>[[1,3,6],[2,5],[4]] [[0,0,1],[1,0,0],[0,1,0]]=>[[1,1,3],[2,3],[3]]=>[4,3,5,1,2,6]=>[[1,3,6],[2,5],[4]] [[0,1,0],[0,0,1],[1,0,0]]=>[[1,2,2],[2,3],[3]]=>[5,2,6,1,3,4]=>[[1,3,6],[2,5],[4]] [[0,0,1],[0,1,0],[1,0,0]]=>[[1,2,3],[2,3],[3]]=>[4,2,5,1,3,6]=>[[1,3,6],[2,5],[4]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]] [[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]=>[[1,3,6,10],[2,5,9],[4,8],[7]]
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
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.