Identifier
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00066: Permutations inversePermutations
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[1]=>[[1]]=>[1]=>[1] [2]=>[[1,2]]=>[1,2]=>[1,2] [1,1]=>[[1],[2]]=>[2,1]=>[2,1] [3]=>[[1,2,3]]=>[1,2,3]=>[1,2,3] [2,1]=>[[1,3],[2]]=>[2,1,3]=>[2,1,3] [1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>[3,2,1] [4]=>[[1,2,3,4]]=>[1,2,3,4]=>[1,2,3,4] [3,1]=>[[1,3,4],[2]]=>[2,1,3,4]=>[2,1,3,4] [2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[3,4,1,2] [2,1,1]=>[[1,4],[2],[3]]=>[3,2,1,4]=>[3,2,1,4] [1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[4,3,2,1] [5]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>[1,2,3,4,5] [4,1]=>[[1,3,4,5],[2]]=>[2,1,3,4,5]=>[2,1,3,4,5] [3,2]=>[[1,2,5],[3,4]]=>[3,4,1,2,5]=>[3,4,1,2,5] [3,1,1]=>[[1,4,5],[2],[3]]=>[3,2,1,4,5]=>[3,2,1,4,5] [2,2,1]=>[[1,3],[2,5],[4]]=>[4,2,5,1,3]=>[4,2,5,1,3] [2,1,1,1]=>[[1,5],[2],[3],[4]]=>[4,3,2,1,5]=>[4,3,2,1,5] [1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[5,4,3,2,1] [6]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6] [5,1]=>[[1,3,4,5,6],[2]]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6] [4,2]=>[[1,2,5,6],[3,4]]=>[3,4,1,2,5,6]=>[3,4,1,2,5,6] [4,1,1]=>[[1,4,5,6],[2],[3]]=>[3,2,1,4,5,6]=>[3,2,1,4,5,6] [3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[4,5,6,1,2,3] [3,2,1]=>[[1,3,6],[2,5],[4]]=>[4,2,5,1,3,6]=>[4,2,5,1,3,6] [3,1,1,1]=>[[1,5,6],[2],[3],[4]]=>[4,3,2,1,5,6]=>[4,3,2,1,5,6] [2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[5,6,3,4,1,2] [2,2,1,1]=>[[1,4],[2,6],[3],[5]]=>[5,3,2,6,1,4]=>[5,3,2,6,1,4] [2,1,1,1,1]=>[[1,6],[2],[3],[4],[5]]=>[5,4,3,2,1,6]=>[5,4,3,2,1,6] [1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1] [7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7] [6,1]=>[[1,3,4,5,6,7],[2]]=>[2,1,3,4,5,6,7]=>[2,1,3,4,5,6,7] [5,2]=>[[1,2,5,6,7],[3,4]]=>[3,4,1,2,5,6,7]=>[3,4,1,2,5,6,7] [5,1,1]=>[[1,4,5,6,7],[2],[3]]=>[3,2,1,4,5,6,7]=>[3,2,1,4,5,6,7] [4,3]=>[[1,2,3,7],[4,5,6]]=>[4,5,6,1,2,3,7]=>[4,5,6,1,2,3,7] [4,2,1]=>[[1,3,6,7],[2,5],[4]]=>[4,2,5,1,3,6,7]=>[4,2,5,1,3,6,7] [4,1,1,1]=>[[1,5,6,7],[2],[3],[4]]=>[4,3,2,1,5,6,7]=>[4,3,2,1,5,6,7] [3,3,1]=>[[1,3,4],[2,6,7],[5]]=>[5,2,6,7,1,3,4]=>[5,2,6,7,1,3,4] [3,2,2]=>[[1,2,7],[3,4],[5,6]]=>[5,6,3,4,1,2,7]=>[5,6,3,4,1,2,7] [3,2,1,1]=>[[1,4,7],[2,6],[3],[5]]=>[5,3,2,6,1,4,7]=>[5,3,2,6,1,4,7] [3,1,1,1,1]=>[[1,6,7],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7]=>[5,4,3,2,1,6,7] [2,2,2,1]=>[[1,3],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3]=>[6,4,7,2,5,1,3] [2,2,1,1,1]=>[[1,5],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5]=>[6,4,3,2,7,1,5] [2,1,1,1,1,1]=>[[1,7],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7]=>[6,5,4,3,2,1,7] [1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[7,6,5,4,3,2,1] [8]=>[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8] [7,1]=>[[1,3,4,5,6,7,8],[2]]=>[2,1,3,4,5,6,7,8]=>[2,1,3,4,5,6,7,8] [6,2]=>[[1,2,5,6,7,8],[3,4]]=>[3,4,1,2,5,6,7,8]=>[3,4,1,2,5,6,7,8] [6,1,1]=>[[1,4,5,6,7,8],[2],[3]]=>[3,2,1,4,5,6,7,8]=>[3,2,1,4,5,6,7,8] [5,3]=>[[1,2,3,7,8],[4,5,6]]=>[4,5,6,1,2,3,7,8]=>[4,5,6,1,2,3,7,8] [5,2,1]=>[[1,3,6,7,8],[2,5],[4]]=>[4,2,5,1,3,6,7,8]=>[4,2,5,1,3,6,7,8] [5,1,1,1]=>[[1,5,6,7,8],[2],[3],[4]]=>[4,3,2,1,5,6,7,8]=>[4,3,2,1,5,6,7,8] [4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[5,6,7,8,1,2,3,4] [4,3,1]=>[[1,3,4,8],[2,6,7],[5]]=>[5,2,6,7,1,3,4,8]=>[5,2,6,7,1,3,4,8] [4,2,2]=>[[1,2,7,8],[3,4],[5,6]]=>[5,6,3,4,1,2,7,8]=>[5,6,3,4,1,2,7,8] [4,2,1,1]=>[[1,4,7,8],[2,6],[3],[5]]=>[5,3,2,6,1,4,7,8]=>[5,3,2,6,1,4,7,8] [4,1,1,1,1]=>[[1,6,7,8],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7,8]=>[5,4,3,2,1,6,7,8] [3,3,2]=>[[1,2,5],[3,4,8],[6,7]]=>[6,7,3,4,8,1,2,5]=>[6,7,3,4,8,1,2,5] [3,3,1,1]=>[[1,4,5],[2,7,8],[3],[6]]=>[6,3,2,7,8,1,4,5]=>[6,3,2,7,8,1,4,5] [3,2,2,1]=>[[1,3,8],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3,8]=>[6,4,7,2,5,1,3,8] [3,2,1,1,1]=>[[1,5,8],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5,8]=>[6,4,3,2,7,1,5,8] [3,1,1,1,1,1]=>[[1,7,8],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7,8]=>[6,5,4,3,2,1,7,8] [2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[7,8,5,6,3,4,1,2] [2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5],[7]]=>[7,5,3,8,2,6,1,4]=>[7,5,3,8,2,6,1,4] [2,2,1,1,1,1]=>[[1,6],[2,8],[3],[4],[5],[7]]=>[7,5,4,3,2,8,1,6]=>[7,5,4,3,2,8,1,6] [2,1,1,1,1,1,1]=>[[1,8],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1,8]=>[7,6,5,4,3,2,1,8] [1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1] [9]=>[[1,2,3,4,5,6,7,8,9]]=>[1,2,3,4,5,6,7,8,9]=>[1,2,3,4,5,6,7,8,9] [8,1]=>[[1,3,4,5,6,7,8,9],[2]]=>[2,1,3,4,5,6,7,8,9]=>[2,1,3,4,5,6,7,8,9] [7,2]=>[[1,2,5,6,7,8,9],[3,4]]=>[3,4,1,2,5,6,7,8,9]=>[3,4,1,2,5,6,7,8,9] [7,1,1]=>[[1,4,5,6,7,8,9],[2],[3]]=>[3,2,1,4,5,6,7,8,9]=>[3,2,1,4,5,6,7,8,9] [6,3]=>[[1,2,3,7,8,9],[4,5,6]]=>[4,5,6,1,2,3,7,8,9]=>[4,5,6,1,2,3,7,8,9] [6,2,1]=>[[1,3,6,7,8,9],[2,5],[4]]=>[4,2,5,1,3,6,7,8,9]=>[4,2,5,1,3,6,7,8,9] [6,1,1,1]=>[[1,5,6,7,8,9],[2],[3],[4]]=>[4,3,2,1,5,6,7,8,9]=>[4,3,2,1,5,6,7,8,9] [5,4]=>[[1,2,3,4,9],[5,6,7,8]]=>[5,6,7,8,1,2,3,4,9]=>[5,6,7,8,1,2,3,4,9] [5,3,1]=>[[1,3,4,8,9],[2,6,7],[5]]=>[5,2,6,7,1,3,4,8,9]=>[5,2,6,7,1,3,4,8,9] [5,2,2]=>[[1,2,7,8,9],[3,4],[5,6]]=>[5,6,3,4,1,2,7,8,9]=>[5,6,3,4,1,2,7,8,9] [5,2,1,1]=>[[1,4,7,8,9],[2,6],[3],[5]]=>[5,3,2,6,1,4,7,8,9]=>[5,3,2,6,1,4,7,8,9] [5,1,1,1,1]=>[[1,6,7,8,9],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7,8,9]=>[5,4,3,2,1,6,7,8,9] [4,4,1]=>[[1,3,4,5],[2,7,8,9],[6]]=>[6,2,7,8,9,1,3,4,5]=>[6,2,7,8,9,1,3,4,5] [4,3,2]=>[[1,2,5,9],[3,4,8],[6,7]]=>[6,7,3,4,8,1,2,5,9]=>[6,7,3,4,8,1,2,5,9] [4,3,1,1]=>[[1,4,5,9],[2,7,8],[3],[6]]=>[6,3,2,7,8,1,4,5,9]=>[6,3,2,7,8,1,4,5,9] [4,2,2,1]=>[[1,3,8,9],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3,8,9]=>[6,4,7,2,5,1,3,8,9] [4,2,1,1,1]=>[[1,5,8,9],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5,8,9]=>[6,4,3,2,7,1,5,8,9] [4,1,1,1,1,1]=>[[1,7,8,9],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7,8,9]=>[6,5,4,3,2,1,7,8,9] [3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[7,8,9,4,5,6,1,2,3]=>[7,8,9,4,5,6,1,2,3] [3,3,2,1]=>[[1,3,6],[2,5,9],[4,8],[7]]=>[7,4,8,2,5,9,1,3,6]=>[7,4,8,2,5,9,1,3,6] [3,3,1,1,1]=>[[1,5,6],[2,8,9],[3],[4],[7]]=>[7,4,3,2,8,9,1,5,6]=>[7,4,3,2,8,9,1,5,6] [3,2,2,2]=>[[1,2,9],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2,9]=>[7,8,5,6,3,4,1,2,9] [3,2,2,1,1]=>[[1,4,9],[2,6],[3,8],[5],[7]]=>[7,5,3,8,2,6,1,4,9]=>[7,5,3,8,2,6,1,4,9] [3,2,1,1,1,1]=>[[1,6,9],[2,8],[3],[4],[5],[7]]=>[7,5,4,3,2,8,1,6,9]=>[7,5,4,3,2,8,1,6,9] [3,1,1,1,1,1,1]=>[[1,8,9],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1,8,9]=>[7,6,5,4,3,2,1,8,9] [2,2,2,2,1]=>[[1,3],[2,5],[4,7],[6,9],[8]]=>[8,6,9,4,7,2,5,1,3]=>[8,6,9,4,7,2,5,1,3] [2,2,2,1,1,1]=>[[1,5],[2,7],[3,9],[4],[6],[8]]=>[8,6,4,3,9,2,7,1,5]=>[8,6,4,3,9,2,7,1,5] [2,2,1,1,1,1,1]=>[[1,7],[2,9],[3],[4],[5],[6],[8]]=>[8,6,5,4,3,2,9,1,7]=>[8,6,5,4,3,2,9,1,7] [2,1,1,1,1,1,1,1]=>[[1,9],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1,9]=>[8,7,6,5,4,3,2,1,9] [1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,4,3,2,1]=>[9,8,7,6,5,4,3,2,1] [10]=>[[1,2,3,4,5,6,7,8,9,10]]=>[1,2,3,4,5,6,7,8,9,10]=>[1,2,3,4,5,6,7,8,9,10] [9,1]=>[[1,3,4,5,6,7,8,9,10],[2]]=>[2,1,3,4,5,6,7,8,9,10]=>[2,1,3,4,5,6,7,8,9,10] [8,2]=>[[1,2,5,6,7,8,9,10],[3,4]]=>[3,4,1,2,5,6,7,8,9,10]=>[3,4,1,2,5,6,7,8,9,10] [8,1,1]=>[[1,4,5,6,7,8,9,10],[2],[3]]=>[3,2,1,4,5,6,7,8,9,10]=>[3,2,1,4,5,6,7,8,9,10] [7,3]=>[[1,2,3,7,8,9,10],[4,5,6]]=>[4,5,6,1,2,3,7,8,9,10]=>[4,5,6,1,2,3,7,8,9,10] [7,2,1]=>[[1,3,6,7,8,9,10],[2,5],[4]]=>[4,2,5,1,3,6,7,8,9,10]=>[4,2,5,1,3,6,7,8,9,10] [7,1,1,1]=>[[1,5,6,7,8,9,10],[2],[3],[4]]=>[4,3,2,1,5,6,7,8,9,10]=>[4,3,2,1,5,6,7,8,9,10] [6,4]=>[[1,2,3,4,9,10],[5,6,7,8]]=>[5,6,7,8,1,2,3,4,9,10]=>[5,6,7,8,1,2,3,4,9,10] [6,3,1]=>[[1,3,4,8,9,10],[2,6,7],[5]]=>[5,2,6,7,1,3,4,8,9,10]=>[5,2,6,7,1,3,4,8,9,10] [6,2,2]=>[[1,2,7,8,9,10],[3,4],[5,6]]=>[5,6,3,4,1,2,7,8,9,10]=>[5,6,3,4,1,2,7,8,9,10] [6,2,1,1]=>[[1,4,7,8,9,10],[2,6],[3],[5]]=>[5,3,2,6,1,4,7,8,9,10]=>[5,3,2,6,1,4,7,8,9,10] [6,1,1,1,1]=>[[1,6,7,8,9,10],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7,8,9,10]=>[5,4,3,2,1,6,7,8,9,10] [5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[6,7,8,9,10,1,2,3,4,5]=>[6,7,8,9,10,1,2,3,4,5] [5,4,1]=>[[1,3,4,5,10],[2,7,8,9],[6]]=>[6,2,7,8,9,1,3,4,5,10]=>[6,2,7,8,9,1,3,4,5,10] [5,3,2]=>[[1,2,5,9,10],[3,4,8],[6,7]]=>[6,7,3,4,8,1,2,5,9,10]=>[6,7,3,4,8,1,2,5,9,10] [5,3,1,1]=>[[1,4,5,9,10],[2,7,8],[3],[6]]=>[6,3,2,7,8,1,4,5,9,10]=>[6,3,2,7,8,1,4,5,9,10] [5,2,2,1]=>[[1,3,8,9,10],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3,8,9,10]=>[6,4,7,2,5,1,3,8,9,10] [5,2,1,1,1]=>[[1,5,8,9,10],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5,8,9,10]=>[6,4,3,2,7,1,5,8,9,10] [5,1,1,1,1,1]=>[[1,7,8,9,10],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7,8,9,10]=>[6,5,4,3,2,1,7,8,9,10] [4,4,2]=>[[1,2,5,6],[3,4,9,10],[7,8]]=>[7,8,3,4,9,10,1,2,5,6]=>[7,8,3,4,9,10,1,2,5,6] [4,4,1,1]=>[[1,4,5,6],[2,8,9,10],[3],[7]]=>[7,3,2,8,9,10,1,4,5,6]=>[7,3,2,8,9,10,1,4,5,6] [4,3,3]=>[[1,2,3,10],[4,5,6],[7,8,9]]=>[7,8,9,4,5,6,1,2,3,10]=>[7,8,9,4,5,6,1,2,3,10] [4,3,2,1]=>[[1,3,6,10],[2,5,9],[4,8],[7]]=>[7,4,8,2,5,9,1,3,6,10]=>[7,4,8,2,5,9,1,3,6,10] [4,3,1,1,1]=>[[1,5,6,10],[2,8,9],[3],[4],[7]]=>[7,4,3,2,8,9,1,5,6,10]=>[7,4,3,2,8,9,1,5,6,10] [4,2,2,2]=>[[1,2,9,10],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2,9,10]=>[7,8,5,6,3,4,1,2,9,10] [4,2,2,1,1]=>[[1,4,9,10],[2,6],[3,8],[5],[7]]=>[7,5,3,8,2,6,1,4,9,10]=>[7,5,3,8,2,6,1,4,9,10] [4,2,1,1,1,1]=>[[1,6,9,10],[2,8],[3],[4],[5],[7]]=>[7,5,4,3,2,8,1,6,9,10]=>[7,5,4,3,2,8,1,6,9,10] [4,1,1,1,1,1,1]=>[[1,8,9,10],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1,8,9,10]=>[7,6,5,4,3,2,1,8,9,10] [3,3,3,1]=>[[1,3,4],[2,6,7],[5,9,10],[8]]=>[8,5,9,10,2,6,7,1,3,4]=>[8,5,9,10,2,6,7,1,3,4] [3,3,2,2]=>[[1,2,7],[3,4,10],[5,6],[8,9]]=>[8,9,5,6,3,4,10,1,2,7]=>[8,9,5,6,3,4,10,1,2,7] [3,3,2,1,1]=>[[1,4,7],[2,6,10],[3,9],[5],[8]]=>[8,5,3,9,2,6,10,1,4,7]=>[8,5,3,9,2,6,10,1,4,7] [3,3,1,1,1,1]=>[[1,6,7],[2,9,10],[3],[4],[5],[8]]=>[8,5,4,3,2,9,10,1,6,7]=>[8,5,4,3,2,9,10,1,6,7] [3,2,2,2,1]=>[[1,3,10],[2,5],[4,7],[6,9],[8]]=>[8,6,9,4,7,2,5,1,3,10]=>[8,6,9,4,7,2,5,1,3,10] [3,2,2,1,1,1]=>[[1,5,10],[2,7],[3,9],[4],[6],[8]]=>[8,6,4,3,9,2,7,1,5,10]=>[8,6,4,3,9,2,7,1,5,10] [3,2,1,1,1,1,1]=>[[1,7,10],[2,9],[3],[4],[5],[6],[8]]=>[8,6,5,4,3,2,9,1,7,10]=>[8,6,5,4,3,2,9,1,7,10] [3,1,1,1,1,1,1,1]=>[[1,9,10],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1,9,10]=>[8,7,6,5,4,3,2,1,9,10] [2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,3,4,1,2]=>[9,10,7,8,5,6,3,4,1,2] [2,2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5,10],[7],[9]]=>[9,7,5,10,3,8,2,6,1,4]=>[9,7,5,10,3,8,2,6,1,4] [2,2,2,1,1,1,1]=>[[1,6],[2,8],[3,10],[4],[5],[7],[9]]=>[9,7,5,4,3,10,2,8,1,6]=>[9,7,5,4,3,10,2,8,1,6] [2,2,1,1,1,1,1,1]=>[[1,8],[2,10],[3],[4],[5],[6],[7],[9]]=>[9,7,6,5,4,3,2,10,1,8]=>[9,7,6,5,4,3,2,10,1,8] [2,1,1,1,1,1,1,1,1]=>[[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,4,3,2,1,10]=>[9,8,7,6,5,4,3,2,1,10] [1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,4,3,2,1]=>[10,9,8,7,6,5,4,3,2,1] [6,6]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>[7,8,9,10,11,12,1,2,3,4,5,6]=>[7,8,9,10,11,12,1,2,3,4,5,6]
Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
inverse
Description
Sends a permutation to its inverse.