Identifier
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
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]=>[1,2] [3]=>[[1,2,3]]=>[1,2,3]=>[1,2,3] [2,1]=>[[1,2],[3]]=>[3,1,2]=>[1,2,3] [1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>[1,2,3] [4]=>[[1,2,3,4]]=>[1,2,3,4]=>[1,2,3,4] [3,1]=>[[1,2,3],[4]]=>[4,1,2,3]=>[1,2,3,4] [2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[1,2,3,4] [2,1,1]=>[[1,2],[3],[4]]=>[4,3,1,2]=>[1,2,3,4] [1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[1,2,3,4] [5]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>[1,2,3,4,5] [4,1]=>[[1,2,3,4],[5]]=>[5,1,2,3,4]=>[1,2,3,4,5] [3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[1,2,3,4,5] [3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[1,2,3,4,5] [2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[1,2,3,4,5] [2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[1,2,3,4,5] [1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[1,2,3,4,5] [6]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6] [5,1]=>[[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[1,2,3,4,5,6] [4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[1,2,3,4,5,6] [4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[1,2,3,4,5,6] [3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[1,2,3,4,5,6] [3,2,1]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[1,2,3,4,5,6] [3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[1,2,3,4,5,6] [2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[1,2,3,4,5,6] [2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[1,2,3,4,5,6] [2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[1,2,3,4,5,6] [1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[1,2,3,4,5,6] [7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7] [6,1]=>[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[1,2,3,4,5,6,7] [5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[1,2,3,4,5,6,7] [5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[1,2,3,4,5,6,7] [4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[1,2,3,4,5,6,7] [4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[1,2,3,4,5,6,7] [4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[1,2,3,4,5,6,7] [3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[1,2,3,4,5,6,7] [3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[1,2,3,4,5,6,7] [3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[1,2,3,4,5,6,7] [3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[1,2,3,4,5,6,7] [2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[1,2,3,4,5,6,7] [2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[1,2,3,4,5,6,7] [2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[1,2,3,4,5,6,7] [1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[1,2,3,4,5,6,7] [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]=>[[1,2,3,4,5,6,7],[8]]=>[8,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7,8] [6,2]=>[[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>[1,2,3,4,5,6,7,8] [6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>[8,7,1,2,3,4,5,6]=>[1,2,3,4,5,6,7,8] [5,3]=>[[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>[1,2,3,4,5,6,7,8] [5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[1,2,3,4,5,6,7,8] [5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[1,2,3,4,5,6,7,8] [4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[1,2,3,4,5,6,7,8] [4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[1,2,3,4,5,6,7,8] [4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[1,2,3,4,5,6,7,8] [4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[1,2,3,4,5,6,7,8] [4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[1,2,3,4,5,6,7,8] [3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>[7,8,4,5,6,1,2,3]=>[1,2,3,4,5,6,7,8] [3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>[8,7,4,5,6,1,2,3]=>[1,2,3,4,5,6,7,8] [3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>[1,2,3,4,5,6,7,8] [3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>[8,7,6,4,5,1,2,3]=>[1,2,3,4,5,6,7,8] [3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>[1,2,3,4,5,6,7,8] [2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[1,2,3,4,5,6,7,8] [2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>[1,2,3,4,5,6,7,8] [2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>[1,2,3,4,5,6,7,8] [2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[1,2,3,4,5,6,7,8] [1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[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]=>[1,2,3,4,5,6,7,8,9] [8,1]=>[[1,2,3,4,5,6,7,8],[9]]=>[9,1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8,9] [7,2]=>[[1,2,3,4,5,6,7],[8,9]]=>[8,9,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7,8,9] [7,1,1]=>[[1,2,3,4,5,6,7],[8],[9]]=>[9,8,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7,8,9] [6,3]=>[[1,2,3,4,5,6],[7,8,9]]=>[7,8,9,1,2,3,4,5,6]=>[1,2,3,4,5,6,7,8,9] [6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>[9,7,8,1,2,3,4,5,6]=>[1,2,3,4,5,6,7,8,9] [6,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9]]=>[9,8,7,1,2,3,4,5,6]=>[1,2,3,4,5,6,7,8,9] [5,4]=>[[1,2,3,4,5],[6,7,8,9]]=>[6,7,8,9,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9] [5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>[9,6,7,8,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9] [5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>[8,9,6,7,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9] [5,2,1,1]=>[[1,2,3,4,5],[6,7],[8],[9]]=>[9,8,6,7,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9] [5,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9]]=>[9,8,7,6,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9] [4,4,1]=>[[1,2,3,4],[5,6,7,8],[9]]=>[9,5,6,7,8,1,2,3,4]=>[1,2,3,4,5,6,7,8,9] [4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>[8,9,5,6,7,1,2,3,4]=>[1,2,3,4,5,6,7,8,9] [4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>[9,8,5,6,7,1,2,3,4]=>[1,2,3,4,5,6,7,8,9] [4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,1,2,3,4]=>[1,2,3,4,5,6,7,8,9] [4,2,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,1,2,3,4]=>[1,2,3,4,5,6,7,8,9] [4,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,1,2,3,4]=>[1,2,3,4,5,6,7,8,9] [3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[7,8,9,4,5,6,1,2,3]=>[1,2,3,4,5,6,7,8,9] [3,3,2,1]=>[[1,2,3],[4,5,6],[7,8],[9]]=>[9,7,8,4,5,6,1,2,3]=>[1,2,3,4,5,6,7,8,9] [3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>[9,8,7,4,5,6,1,2,3]=>[1,2,3,4,5,6,7,8,9] [3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>[8,9,6,7,4,5,1,2,3]=>[1,2,3,4,5,6,7,8,9] [3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>[9,8,6,7,4,5,1,2,3]=>[1,2,3,4,5,6,7,8,9] [3,2,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9]]=>[9,8,7,6,4,5,1,2,3]=>[1,2,3,4,5,6,7,8,9] [3,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,4,1,2,3]=>[1,2,3,4,5,6,7,8,9] [2,2,2,2,1]=>[[1,2],[3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,3,4,1,2]=>[1,2,3,4,5,6,7,8,9] [2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,3,4,1,2]=>[1,2,3,4,5,6,7,8,9] [2,2,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,3,4,1,2]=>[1,2,3,4,5,6,7,8,9] [2,1,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,4,3,1,2]=>[1,2,3,4,5,6,7,8,9] [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]=>[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]=>[1,2,3,4,5,6,7,8,9,10] [9,1]=>[[1,2,3,4,5,6,7,8,9],[10]]=>[10,1,2,3,4,5,6,7,8,9]=>[1,2,3,4,5,6,7,8,9,10] [8,2]=>[[1,2,3,4,5,6,7,8],[9,10]]=>[9,10,1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8,9,10] [8,1,1]=>[[1,2,3,4,5,6,7,8],[9],[10]]=>[10,9,1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8,9,10] [7,3]=>[[1,2,3,4,5,6,7],[8,9,10]]=>[8,9,10,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7,8,9,10] [7,2,1]=>[[1,2,3,4,5,6,7],[8,9],[10]]=>[10,8,9,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7,8,9,10] [7,1,1,1]=>[[1,2,3,4,5,6,7],[8],[9],[10]]=>[10,9,8,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7,8,9,10] [6,4]=>[[1,2,3,4,5,6],[7,8,9,10]]=>[7,8,9,10,1,2,3,4,5,6]=>[1,2,3,4,5,6,7,8,9,10] [6,3,1]=>[[1,2,3,4,5,6],[7,8,9],[10]]=>[10,7,8,9,1,2,3,4,5,6]=>[1,2,3,4,5,6,7,8,9,10] [6,2,2]=>[[1,2,3,4,5,6],[7,8],[9,10]]=>[9,10,7,8,1,2,3,4,5,6]=>[1,2,3,4,5,6,7,8,9,10] [6,2,1,1]=>[[1,2,3,4,5,6],[7,8],[9],[10]]=>[10,9,7,8,1,2,3,4,5,6]=>[1,2,3,4,5,6,7,8,9,10] [6,1,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9],[10]]=>[10,9,8,7,1,2,3,4,5,6]=>[1,2,3,4,5,6,7,8,9,10] [5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[6,7,8,9,10,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9,10] [5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>[10,6,7,8,9,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9,10] [5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>[9,10,6,7,8,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9,10] [5,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10]]=>[10,9,6,7,8,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9,10] [5,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10]]=>[10,8,9,6,7,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9,10] [5,2,1,1,1]=>[[1,2,3,4,5],[6,7],[8],[9],[10]]=>[10,9,8,6,7,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9,10] [5,1,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,1,2,3,4,5]=>[1,2,3,4,5,6,7,8,9,10] [4,4,2]=>[[1,2,3,4],[5,6,7,8],[9,10]]=>[9,10,5,6,7,8,1,2,3,4]=>[1,2,3,4,5,6,7,8,9,10] [4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>[10,9,5,6,7,8,1,2,3,4]=>[1,2,3,4,5,6,7,8,9,10] [4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>[8,9,10,5,6,7,1,2,3,4]=>[1,2,3,4,5,6,7,8,9,10] [4,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10]]=>[10,8,9,5,6,7,1,2,3,4]=>[1,2,3,4,5,6,7,8,9,10] [4,3,1,1,1]=>[[1,2,3,4],[5,6,7],[8],[9],[10]]=>[10,9,8,5,6,7,1,2,3,4]=>[1,2,3,4,5,6,7,8,9,10] [4,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,1,2,3,4]=>[1,2,3,4,5,6,7,8,9,10] [4,2,2,1,1]=>[[1,2,3,4],[5,6],[7,8],[9],[10]]=>[10,9,7,8,5,6,1,2,3,4]=>[1,2,3,4,5,6,7,8,9,10] [4,2,1,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9],[10]]=>[10,9,8,7,5,6,1,2,3,4]=>[1,2,3,4,5,6,7,8,9,10] [4,1,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,1,2,3,4]=>[1,2,3,4,5,6,7,8,9,10] [3,3,3,1]=>[[1,2,3],[4,5,6],[7,8,9],[10]]=>[10,7,8,9,4,5,6,1,2,3]=>[1,2,3,4,5,6,7,8,9,10] [3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>[9,10,7,8,4,5,6,1,2,3]=>[1,2,3,4,5,6,7,8,9,10] [3,3,2,1,1]=>[[1,2,3],[4,5,6],[7,8],[9],[10]]=>[10,9,7,8,4,5,6,1,2,3]=>[1,2,3,4,5,6,7,8,9,10] [3,3,1,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9],[10]]=>[10,9,8,7,4,5,6,1,2,3]=>[1,2,3,4,5,6,7,8,9,10] [3,2,2,2,1]=>[[1,2,3],[4,5],[6,7],[8,9],[10]]=>[10,8,9,6,7,4,5,1,2,3]=>[1,2,3,4,5,6,7,8,9,10] [3,2,2,1,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9],[10]]=>[10,9,8,6,7,4,5,1,2,3]=>[1,2,3,4,5,6,7,8,9,10] [3,2,1,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,4,5,1,2,3]=>[1,2,3,4,5,6,7,8,9,10] [3,1,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,4,1,2,3]=>[1,2,3,4,5,6,7,8,9,10] [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]=>[1,2,3,4,5,6,7,8,9,10] [2,2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7,8],[9],[10]]=>[10,9,7,8,5,6,3,4,1,2]=>[1,2,3,4,5,6,7,8,9,10] [2,2,2,1,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9],[10]]=>[10,9,8,7,5,6,3,4,1,2]=>[1,2,3,4,5,6,7,8,9,10] [2,2,1,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,3,4,1,2]=>[1,2,3,4,5,6,7,8,9,10] [2,1,1,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,4,3,1,2]=>[1,2,3,4,5,6,7,8,9,10] [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]=>[1,2,3,4,5,6,7,8,9,10] [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]=>[1,2,3,4,5,6,7,8,9,10,11,12] [1,1,1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]=>[12,11,10,9,8,7,6,5,4,3,2,1]=>[1,2,3,4,5,6,7,8,9,10,11,12]
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.