Identifier
Mp00001: to semistandard tableau via monotone trianglesSemistandard tableaux
Mp00064: Permutations reversePermutations
Mp00070: Permutations Robinson-Schensted recording tableau
Images
=>
Cc0017;cc-rep-0Cc0019;cc-rep-1Cc0007;cc-rep-4
[[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
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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.