Identifier
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Images
=>
Cc0017;cc-rep-1Cc0019;cc-rep-2Cc0002;cc-rep-3
[1]=>[[1]]=>[[1]]=>[1] [1,2]=>[[1,0],[0,1]]=>[[1,1],[2]]=>[2,1] [2,1]=>[[0,1],[1,0]]=>[[1,2],[2]]=>[2,1] [1,2,3]=>[[1,0,0],[0,1,0],[0,0,1]]=>[[1,1,1],[2,2],[3]]=>[3,2,1] [1,3,2]=>[[1,0,0],[0,0,1],[0,1,0]]=>[[1,1,1],[2,3],[3]]=>[3,2,1] [2,1,3]=>[[0,1,0],[1,0,0],[0,0,1]]=>[[1,1,2],[2,2],[3]]=>[3,2,1] [2,3,1]=>[[0,0,1],[1,0,0],[0,1,0]]=>[[1,1,3],[2,3],[3]]=>[3,2,1] [3,1,2]=>[[0,1,0],[0,0,1],[1,0,0]]=>[[1,2,2],[2,3],[3]]=>[3,2,1] [3,2,1]=>[[0,0,1],[0,1,0],[1,0,0]]=>[[1,2,3],[2,3],[3]]=>[3,2,1] [1,2,3,4]=>[[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]]=>[4,3,2,1] [1,2,4,3]=>[[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]]=>[4,3,2,1] [1,3,2,4]=>[[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]]=>[4,3,2,1] [1,3,4,2]=>[[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]]=>[4,3,2,1] [1,4,2,3]=>[[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]]=>[4,3,2,1] [1,4,3,2]=>[[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]]=>[4,3,2,1] [2,1,3,4]=>[[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]]=>[4,3,2,1] [2,1,4,3]=>[[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]]=>[4,3,2,1] [2,3,1,4]=>[[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]]=>[4,3,2,1] [2,3,4,1]=>[[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]]=>[4,3,2,1] [2,4,1,3]=>[[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]]=>[4,3,2,1] [2,4,3,1]=>[[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]]=>[4,3,2,1] [3,1,2,4]=>[[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]]=>[4,3,2,1] [3,1,4,2]=>[[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]]=>[4,3,2,1] [3,2,1,4]=>[[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]]=>[4,3,2,1] [3,2,4,1]=>[[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]]=>[4,3,2,1] [3,4,1,2]=>[[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]]=>[4,3,2,1] [3,4,2,1]=>[[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]]=>[4,3,2,1] [4,1,2,3]=>[[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]]=>[4,3,2,1] [4,1,3,2]=>[[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]]=>[4,3,2,1] [4,2,1,3]=>[[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]]=>[4,3,2,1] [4,2,3,1]=>[[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]]=>[4,3,2,1] [4,3,1,2]=>[[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]]=>[4,3,2,1] [4,3,2,1]=>[[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]]=>[4,3,2,1] [1,2,3,4,5]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]=>[5,4,3,2,1] [1,2,3,5,4]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]=>[5,4,3,2,1] [1,2,4,3,5]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [1,2,4,5,3]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [1,2,5,3,4]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]=>[[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [1,2,5,4,3]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]=>[[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [1,3,2,4,5]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]=>[5,4,3,2,1] [1,3,2,5,4]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]=>[5,4,3,2,1] [1,3,4,2,5]=>[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [1,3,4,5,2]=>[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [1,3,5,2,4]=>[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]=>[[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [1,3,5,4,2]=>[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]=>[[1,1,1,1,1],[2,2,2,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [1,4,2,3,5]=>[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]=>[[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [1,4,2,5,3]=>[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]=>[[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [1,4,3,2,5]=>[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]=>[[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [1,4,3,5,2]=>[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]=>[[1,1,1,1,1],[2,2,3,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [1,4,5,2,3]=>[[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]=>[[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [1,4,5,3,2]=>[[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]=>[[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [1,5,2,3,4]=>[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]=>[[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [1,5,2,4,3]=>[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]=>[[1,1,1,1,1],[2,3,3,3],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [1,5,3,2,4]=>[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]=>[[1,1,1,1,1],[2,3,3,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [1,5,3,4,2]=>[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]=>[[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [1,5,4,2,3]=>[[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]=>[[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [1,5,4,3,2]=>[[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]=>[[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [2,1,3,4,5]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]=>[5,4,3,2,1] [2,1,3,5,4]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]=>[5,4,3,2,1] [2,1,4,3,5]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [2,1,4,5,3]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [2,1,5,3,4]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]=>[[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [2,1,5,4,3]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]=>[[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [2,3,1,4,5]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]=>[5,4,3,2,1] [2,3,1,5,4]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]=>[5,4,3,2,1] [2,3,4,1,5]=>[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [2,3,4,5,1]=>[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [2,3,5,1,4]=>[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]=>[[1,1,1,1,4],[2,2,2,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [2,3,5,4,1]=>[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]=>[[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [2,4,1,3,5]=>[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]=>[[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [2,4,1,5,3]=>[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]=>[[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [2,4,3,1,5]=>[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]=>[[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [2,4,3,5,1]=>[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]=>[[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [2,4,5,1,3]=>[[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]=>[[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [2,4,5,3,1]=>[[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]=>[[1,1,1,1,5],[2,2,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [2,5,1,3,4]=>[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]=>[[1,1,1,1,3],[2,3,3,3],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [2,5,1,4,3]=>[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]=>[[1,1,1,1,3],[2,3,3,3],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [2,5,3,1,4]=>[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]=>[[1,1,1,1,4],[2,3,3,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [2,5,3,4,1]=>[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]=>[[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [2,5,4,1,3]=>[[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]=>[[1,1,1,1,4],[2,3,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [2,5,4,3,1]=>[[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]=>[[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [3,1,2,4,5]=>[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]=>[5,4,3,2,1] [3,1,2,5,4]=>[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]=>[5,4,3,2,1] [3,1,4,2,5]=>[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [3,1,4,5,2]=>[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[[1,1,1,2,2],[2,2,2,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [3,1,5,2,4]=>[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]=>[[1,1,1,2,2],[2,2,2,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [3,1,5,4,2]=>[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]=>[[1,1,1,2,2],[2,2,2,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [3,2,1,4,5]=>[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]=>[5,4,3,2,1] [3,2,1,5,4]=>[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]=>[5,4,3,2,1] [3,2,4,1,5]=>[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [3,2,4,5,1]=>[[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[[1,1,1,2,5],[2,2,2,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [3,2,5,1,4]=>[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]=>[[1,1,1,2,4],[2,2,2,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [3,2,5,4,1]=>[[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]=>[[1,1,1,2,5],[2,2,2,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [3,4,1,2,5]=>[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]=>[[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [3,4,1,5,2]=>[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]=>[[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [3,4,2,1,5]=>[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]=>[[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [3,4,2,5,1]=>[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]=>[[1,1,1,3,5],[2,2,3,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [3,4,5,1,2]=>[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]=>[[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [3,4,5,2,1]=>[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]=>[[1,1,1,4,5],[2,2,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [3,5,1,2,4]=>[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]=>[[1,1,1,3,3],[2,3,3,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [3,5,1,4,2]=>[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]=>[[1,1,1,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [3,5,2,1,4]=>[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]=>[[1,1,1,3,4],[2,3,3,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [3,5,2,4,1]=>[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]=>[[1,1,1,3,5],[2,3,3,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [3,5,4,1,2]=>[[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]=>[[1,1,1,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [3,5,4,2,1]=>[[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]=>[[1,1,1,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,1,2,3,5]=>[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]=>[[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [4,1,2,5,3]=>[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]=>[[1,1,2,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [4,1,3,2,5]=>[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]=>[[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [4,1,3,5,2]=>[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]=>[[1,1,2,2,2],[2,2,3,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [4,1,5,2,3]=>[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]=>[[1,1,2,2,2],[2,2,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,1,5,3,2]=>[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]=>[[1,1,2,2,2],[2,2,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,2,1,3,5]=>[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]=>[[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [4,2,1,5,3]=>[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]=>[[1,1,2,2,3],[2,2,3,3],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [4,2,3,1,5]=>[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]=>[[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [4,2,3,5,1]=>[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]=>[[1,1,2,2,5],[2,2,3,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [4,2,5,1,3]=>[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]=>[[1,1,2,2,4],[2,2,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,2,5,3,1]=>[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]=>[[1,1,2,2,5],[2,2,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,3,1,2,5]=>[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]=>[[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [4,3,1,5,2]=>[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]=>[[1,1,2,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [4,3,2,1,5]=>[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]=>[[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]=>[5,4,3,2,1] [4,3,2,5,1]=>[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]=>[[1,1,2,3,5],[2,2,3,5],[3,3,5],[4,5],[5]]=>[5,4,3,2,1] [4,3,5,1,2]=>[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]=>[[1,1,2,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,3,5,2,1]=>[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]=>[[1,1,2,4,5],[2,2,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,5,1,2,3]=>[[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]=>[[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,5,1,3,2]=>[[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]=>[[1,1,3,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,5,2,1,3]=>[[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]=>[[1,1,3,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,5,2,3,1]=>[[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]=>[[1,1,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,5,3,1,2]=>[[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]=>[[1,1,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [4,5,3,2,1]=>[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]=>[[1,1,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,1,2,3,4]=>[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]=>[[1,2,2,2,2],[2,3,3,3],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [5,1,2,4,3]=>[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]=>[[1,2,2,2,2],[2,3,3,3],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,1,3,2,4]=>[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]=>[[1,2,2,2,2],[2,3,3,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [5,1,3,4,2]=>[[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]=>[[1,2,2,2,2],[2,3,3,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,1,4,2,3]=>[[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]=>[[1,2,2,2,2],[2,3,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,1,4,3,2]=>[[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]=>[[1,2,2,2,2],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,2,1,3,4]=>[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]=>[[1,2,2,2,3],[2,3,3,3],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [5,2,1,4,3]=>[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]=>[[1,2,2,2,3],[2,3,3,3],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,2,3,1,4]=>[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]=>[[1,2,2,2,4],[2,3,3,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [5,2,3,4,1]=>[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]=>[[1,2,2,2,5],[2,3,3,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,2,4,1,3]=>[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]=>[[1,2,2,2,4],[2,3,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,2,4,3,1]=>[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]=>[[1,2,2,2,5],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,3,1,2,4]=>[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]=>[[1,2,2,3,3],[2,3,3,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [5,3,1,4,2]=>[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]=>[[1,2,2,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,3,2,1,4]=>[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]=>[[1,2,2,3,4],[2,3,3,4],[3,4,4],[4,5],[5]]=>[5,4,3,2,1] [5,3,2,4,1]=>[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]=>[[1,2,2,3,5],[2,3,3,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,3,4,1,2]=>[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]=>[[1,2,2,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,3,4,2,1]=>[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]=>[[1,2,2,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,4,1,2,3]=>[[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]=>[[1,2,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,4,1,3,2]=>[[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]=>[[1,2,3,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,4,2,1,3]=>[[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]=>[[1,2,3,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,4,2,3,1]=>[[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]=>[[1,2,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,4,3,1,2]=>[[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]=>[[1,2,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [5,4,3,2,1]=>[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]=>[[1,2,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]=>[5,4,3,2,1] [1,2,3,4,5,6]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [1,2,3,4,6,5]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [1,2,3,5,4,6]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [1,2,3,5,6,4]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]=>[6,5,4,3,2,1] [1,2,3,6,5,4]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]=>[6,5,4,3,2,1] [1,2,4,3,5,6]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [1,2,4,3,6,5]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [1,2,4,5,3,6]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [1,2,4,5,6,3]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]=>[6,5,4,3,2,1] [1,2,5,4,3,6]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [1,2,6,5,4,3]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]=>[[1,1,1,1,1,1],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]=>[6,5,4,3,2,1] [1,3,2,4,5,6]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [1,3,2,4,6,5]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [1,3,2,5,4,6]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [1,3,2,5,6,4]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]=>[6,5,4,3,2,1] [1,3,2,6,5,4]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]=>[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]=>[6,5,4,3,2,1] [1,3,4,2,5,6]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [1,3,4,2,6,5]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [1,3,4,5,2,6]=>[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [1,3,4,5,6,2]=>[[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[[1,1,1,1,1,1],[2,2,2,2,6],[3,3,3,6],[4,4,6],[5,6],[6]]=>[6,5,4,3,2,1] [1,4,3,2,5,6]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [1,4,3,2,6,5]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [1,5,4,3,2,6]=>[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [1,6,5,4,3,2]=>[[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]=>[[1,1,1,1,1,1],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]]=>[6,5,4,3,2,1] [2,1,3,4,5,6]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [2,1,3,4,6,5]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [2,1,3,5,4,6]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [2,1,3,5,6,4]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]=>[6,5,4,3,2,1] [2,1,3,6,5,4]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]=>[6,5,4,3,2,1] [2,1,4,3,5,6]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [2,1,4,3,6,5]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [2,1,4,5,3,6]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [2,1,4,5,6,3]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]=>[6,5,4,3,2,1] [2,1,5,4,3,6]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [2,1,6,5,4,3]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]=>[[1,1,1,1,1,2],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]]=>[6,5,4,3,2,1] [2,3,1,4,5,6]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [2,3,1,4,6,5]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [2,3,1,5,4,6]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [2,3,1,5,6,4]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]=>[6,5,4,3,2,1] [2,3,4,1,5,6]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [2,3,4,1,6,5]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [2,3,4,5,1,6]=>[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,1,5],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [2,3,4,5,6,1]=>[[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[[1,1,1,1,1,6],[2,2,2,2,6],[3,3,3,6],[4,4,6],[5,6],[6]]=>[6,5,4,3,2,1] [3,2,1,4,5,6]=>[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [3,2,1,4,6,5]=>[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [3,2,1,5,4,6]=>[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [3,2,1,6,5,4]=>[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]=>[[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]=>[6,5,4,3,2,1] [4,3,2,1,5,6]=>[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[[1,1,1,2,3,4],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]=>[6,5,4,3,2,1] [4,3,2,1,6,5]=>[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[[1,1,1,2,3,4],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]]=>[6,5,4,3,2,1] [5,4,3,2,1,6]=>[[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]=>[[1,1,2,3,4,5],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]=>[6,5,4,3,2,1] [6,5,4,3,2,1]=>[[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]=>[[1,2,3,4,5,6],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]]=>[6,5,4,3,2,1]
Map
to alternating sign matrix
Description
Maps a permutation to its permutation matrix as an alternating sign matrix.
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
weight
Description
The weight of a semistandard tableau as an integer partition.
The weight (or content) of a semistandard tableaux $T$ with maximal entry $m$ is the weak composition $(\alpha_1, \dots, \alpha_m)$ such that $\alpha_i$ is the number of letters $i$ occurring in $T$.
This map returns the integer partition obtained by sorting the weight into decreasing order and omitting zeros.
Since semistandard tableaux are bigraded by the size of the partition and the maximal occurring entry, this map is not graded.