Identifier
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1Cc0007;cc-rep-4
[1]=>[[1]]=>[1]=>[1]=>[[1]] [2]=>[[1,2]]=>[1,2]=>[2,1]=>[[1],[2]] [1,1]=>[[1],[2]]=>[2,1]=>[1,2]=>[[1,2]] [3]=>[[1,2,3]]=>[1,2,3]=>[3,2,1]=>[[1],[2],[3]] [2,1]=>[[1,2],[3]]=>[3,1,2]=>[1,3,2]=>[[1,2],[3]] [1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>[1,2,3]=>[[1,2,3]] [4]=>[[1,2,3,4]]=>[1,2,3,4]=>[4,3,2,1]=>[[1],[2],[3],[4]] [3,1]=>[[1,2,3],[4]]=>[4,1,2,3]=>[1,4,3,2]=>[[1,2],[3],[4]] [2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,1,4,3]=>[[1,3],[2,4]] [2,1,1]=>[[1,2],[3],[4]]=>[4,3,1,2]=>[1,2,4,3]=>[[1,2,3],[4]] [1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[1,2,3,4]=>[[1,2,3,4]] [5]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>[5,4,3,2,1]=>[[1],[2],[3],[4],[5]] [4,1]=>[[1,2,3,4],[5]]=>[5,1,2,3,4]=>[1,5,4,3,2]=>[[1,2],[3],[4],[5]] [3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[2,1,5,4,3]=>[[1,3],[2,4],[5]] [3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[1,2,5,4,3]=>[[1,2,3],[4],[5]] [2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[1,3,2,5,4]=>[[1,2,4],[3,5]] [2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[1,2,3,5,4]=>[[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]=>[[1,2,3,4,5]] [6]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[6,5,4,3,2,1]=>[[1],[2],[3],[4],[5],[6]] [5,1]=>[[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[1,6,5,4,3,2]=>[[1,2],[3],[4],[5],[6]] [4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[2,1,6,5,4,3]=>[[1,3],[2,4],[5],[6]] [4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[1,2,6,5,4,3]=>[[1,2,3],[4],[5],[6]] [3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[3,2,1,6,5,4]=>[[1,4],[2,5],[3,6]] [3,2,1]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[1,3,2,6,5,4]=>[[1,2,4],[3,5],[6]] [3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[1,2,3,6,5,4]=>[[1,2,3,4],[5],[6]] [2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[2,1,4,3,6,5]=>[[1,3,5],[2,4,6]] [2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[1,2,4,3,6,5]=>[[1,2,3,5],[4,6]] [2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[1,2,3,4,6,5]=>[[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]=>[[1,2,3,4,5,6]] [7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[7,6,5,4,3,2,1]=>[[1],[2],[3],[4],[5],[6],[7]] [6,1]=>[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[1,7,6,5,4,3,2]=>[[1,2],[3],[4],[5],[6],[7]] [5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[2,1,7,6,5,4,3]=>[[1,3],[2,4],[5],[6],[7]] [5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[1,2,7,6,5,4,3]=>[[1,2,3],[4],[5],[6],[7]] [4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[3,2,1,7,6,5,4]=>[[1,4],[2,5],[3,6],[7]] [4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[1,3,2,7,6,5,4]=>[[1,2,4],[3,5],[6],[7]] [4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[1,2,3,7,6,5,4]=>[[1,2,3,4],[5],[6],[7]] [3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[1,4,3,2,7,6,5]=>[[1,2,5],[3,6],[4,7]] [3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[2,1,4,3,7,6,5]=>[[1,3,5],[2,4,6],[7]] [3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[1,2,4,3,7,6,5]=>[[1,2,3,5],[4,6],[7]] [3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[1,2,3,4,7,6,5]=>[[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]=>[1,3,2,5,4,7,6]=>[[1,2,4,6],[3,5,7]] [2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[1,2,3,5,4,7,6]=>[[1,2,3,4,6],[5,7]] [2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[1,2,3,4,5,7,6]=>[[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]=>[[1,2,3,4,5,6,7]] [8]=>[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[8,7,6,5,4,3,2,1]=>[[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]=>[1,8,7,6,5,4,3,2]=>[[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]=>[2,1,8,7,6,5,4,3]=>[[1,3],[2,4],[5],[6],[7],[8]] [6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>[8,7,1,2,3,4,5,6]=>[1,2,8,7,6,5,4,3]=>[[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]=>[3,2,1,8,7,6,5,4]=>[[1,4],[2,5],[3,6],[7],[8]] [5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[1,3,2,8,7,6,5,4]=>[[1,2,4],[3,5],[6],[7],[8]] [5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[1,2,3,8,7,6,5,4]=>[[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]=>[4,3,2,1,8,7,6,5]=>[[1,5],[2,6],[3,7],[4,8]] [4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[1,4,3,2,8,7,6,5]=>[[1,2,5],[3,6],[4,7],[8]] [4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[2,1,4,3,8,7,6,5]=>[[1,3,5],[2,4,6],[7],[8]] [4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[1,2,4,3,8,7,6,5]=>[[1,2,3,5],[4,6],[7],[8]] [4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[1,2,3,4,8,7,6,5]=>[[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]=>[2,1,5,4,3,8,7,6]=>[[1,3,6],[2,4,7],[5,8]] [3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>[8,7,4,5,6,1,2,3]=>[1,2,5,4,3,8,7,6]=>[[1,2,3,6],[4,7],[5,8]] [3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>[1,3,2,5,4,8,7,6]=>[[1,2,4,6],[3,5,7],[8]] [3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>[8,7,6,4,5,1,2,3]=>[1,2,3,5,4,8,7,6]=>[[1,2,3,4,6],[5,7],[8]] [3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>[1,2,3,4,5,8,7,6]=>[[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]=>[2,1,4,3,6,5,8,7]=>[[1,3,5,7],[2,4,6,8]] [2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>[1,2,4,3,6,5,8,7]=>[[1,2,3,5,7],[4,6,8]] [2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>[1,2,3,4,6,5,8,7]=>[[1,2,3,4,5,7],[6,8]] [2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[1,2,3,4,5,6,8,7]=>[[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]=>[[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]=>[9,8,7,6,5,4,3,2,1]=>[[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]=>[1,9,8,7,6,5,4,3,2]=>[[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]=>[2,1,9,8,7,6,5,4,3]=>[[1,3],[2,4],[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]=>[1,2,9,8,7,6,5,4,3]=>[[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]=>[3,2,1,9,8,7,6,5,4]=>[[1,4],[2,5],[3,6],[7],[8],[9]] [6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>[9,7,8,1,2,3,4,5,6]=>[1,3,2,9,8,7,6,5,4]=>[[1,2,4],[3,5],[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]=>[1,2,3,9,8,7,6,5,4]=>[[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]=>[4,3,2,1,9,8,7,6,5]=>[[1,5],[2,6],[3,7],[4,8],[9]] [5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>[9,6,7,8,1,2,3,4,5]=>[1,4,3,2,9,8,7,6,5]=>[[1,2,5],[3,6],[4,7],[8],[9]] [5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>[8,9,6,7,1,2,3,4,5]=>[2,1,4,3,9,8,7,6,5]=>[[1,3,5],[2,4,6],[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]=>[1,2,4,3,9,8,7,6,5]=>[[1,2,3,5],[4,6],[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]=>[1,2,3,4,9,8,7,6,5]=>[[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]=>[1,5,4,3,2,9,8,7,6]=>[[1,2,6],[3,7],[4,8],[5,9]] [4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>[8,9,5,6,7,1,2,3,4]=>[2,1,5,4,3,9,8,7,6]=>[[1,3,6],[2,4,7],[5,8],[9]] [4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>[9,8,5,6,7,1,2,3,4]=>[1,2,5,4,3,9,8,7,6]=>[[1,2,3,6],[4,7],[5,8],[9]] [4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,1,2,3,4]=>[1,3,2,5,4,9,8,7,6]=>[[1,2,4,6],[3,5,7],[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]=>[1,2,3,5,4,9,8,7,6]=>[[1,2,3,4,6],[5,7],[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]=>[1,2,3,4,5,9,8,7,6]=>[[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]=>[3,2,1,6,5,4,9,8,7]=>[[1,4,7],[2,5,8],[3,6,9]] [3,3,2,1]=>[[1,2,3],[4,5,6],[7,8],[9]]=>[9,7,8,4,5,6,1,2,3]=>[1,3,2,6,5,4,9,8,7]=>[[1,2,4,7],[3,5,8],[6,9]] [3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>[9,8,7,4,5,6,1,2,3]=>[1,2,3,6,5,4,9,8,7]=>[[1,2,3,4,7],[5,8],[6,9]] [3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>[8,9,6,7,4,5,1,2,3]=>[2,1,4,3,6,5,9,8,7]=>[[1,3,5,7],[2,4,6,8],[9]] [3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>[9,8,6,7,4,5,1,2,3]=>[1,2,4,3,6,5,9,8,7]=>[[1,2,3,5,7],[4,6,8],[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]=>[1,2,3,4,6,5,9,8,7]=>[[1,2,3,4,5,7],[6,8],[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]=>[1,2,3,4,5,6,9,8,7]=>[[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]=>[1,3,2,5,4,7,6,9,8]=>[[1,2,4,6,8],[3,5,7,9]] [2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,3,4,1,2]=>[1,2,3,5,4,7,6,9,8]=>[[1,2,3,4,6,8],[5,7,9]] [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]=>[1,2,3,4,5,7,6,9,8]=>[[1,2,3,4,5,6,8],[7,9]] [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]=>[1,2,3,4,5,6,7,9,8]=>[[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]=>[[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]=>[10,9,8,7,6,5,4,3,2,1]=>[[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]=>[1,10,9,8,7,6,5,4,3,2]=>[[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]=>[2,1,10,9,8,7,6,5,4,3]=>[[1,3],[2,4],[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]=>[1,2,10,9,8,7,6,5,4,3]=>[[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]=>[3,2,1,10,9,8,7,6,5,4]=>[[1,4],[2,5],[3,6],[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]=>[1,3,2,10,9,8,7,6,5,4]=>[[1,2,4],[3,5],[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]=>[1,2,3,10,9,8,7,6,5,4]=>[[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]=>[4,3,2,1,10,9,8,7,6,5]=>[[1,5],[2,6],[3,7],[4,8],[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]=>[1,4,3,2,10,9,8,7,6,5]=>[[1,2,5],[3,6],[4,7],[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]=>[2,1,4,3,10,9,8,7,6,5]=>[[1,3,5],[2,4,6],[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]=>[1,2,4,3,10,9,8,7,6,5]=>[[1,2,3,5],[4,6],[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]=>[1,2,3,4,10,9,8,7,6,5]=>[[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]=>[5,4,3,2,1,10,9,8,7,6]=>[[1,6],[2,7],[3,8],[4,9],[5,10]] [5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>[10,6,7,8,9,1,2,3,4,5]=>[1,5,4,3,2,10,9,8,7,6]=>[[1,2,6],[3,7],[4,8],[5,9],[10]] [5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>[9,10,6,7,8,1,2,3,4,5]=>[2,1,5,4,3,10,9,8,7,6]=>[[1,3,6],[2,4,7],[5,8],[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]=>[1,2,5,4,3,10,9,8,7,6]=>[[1,2,3,6],[4,7],[5,8],[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]=>[1,3,2,5,4,10,9,8,7,6]=>[[1,2,4,6],[3,5,7],[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]=>[1,2,3,5,4,10,9,8,7,6]=>[[1,2,3,4,6],[5,7],[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]=>[1,2,3,4,5,10,9,8,7,6]=>[[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]=>[2,1,6,5,4,3,10,9,8,7]=>[[1,3,7],[2,4,8],[5,9],[6,10]] [4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>[10,9,5,6,7,8,1,2,3,4]=>[1,2,6,5,4,3,10,9,8,7]=>[[1,2,3,7],[4,8],[5,9],[6,10]] [4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>[8,9,10,5,6,7,1,2,3,4]=>[3,2,1,6,5,4,10,9,8,7]=>[[1,4,7],[2,5,8],[3,6,9],[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]=>[1,3,2,6,5,4,10,9,8,7]=>[[1,2,4,7],[3,5,8],[6,9],[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]=>[1,2,3,6,5,4,10,9,8,7]=>[[1,2,3,4,7],[5,8],[6,9],[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]=>[2,1,4,3,6,5,10,9,8,7]=>[[1,3,5,7],[2,4,6,8],[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]=>[1,2,4,3,6,5,10,9,8,7]=>[[1,2,3,5,7],[4,6,8],[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]=>[1,2,3,4,6,5,10,9,8,7]=>[[1,2,3,4,5,7],[6,8],[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]=>[1,2,3,4,5,6,10,9,8,7]=>[[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]=>[1,4,3,2,7,6,5,10,9,8]=>[[1,2,5,8],[3,6,9],[4,7,10]] [3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>[9,10,7,8,4,5,6,1,2,3]=>[2,1,4,3,7,6,5,10,9,8]=>[[1,3,5,8],[2,4,6,9],[7,10]] [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]=>[1,2,4,3,7,6,5,10,9,8]=>[[1,2,3,5,8],[4,6,9],[7,10]] [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]=>[1,2,3,4,7,6,5,10,9,8]=>[[1,2,3,4,5,8],[6,9],[7,10]] [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]=>[1,3,2,5,4,7,6,10,9,8]=>[[1,2,4,6,8],[3,5,7,9],[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]=>[1,2,3,5,4,7,6,10,9,8]=>[[1,2,3,4,6,8],[5,7,9],[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]=>[1,2,3,4,5,7,6,10,9,8]=>[[1,2,3,4,5,6,8],[7,9],[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]=>[1,2,3,4,5,6,7,10,9,8]=>[[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]=>[2,1,4,3,6,5,8,7,10,9]=>[[1,3,5,7,9],[2,4,6,8,10]] [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]=>[1,2,4,3,6,5,8,7,10,9]=>[[1,2,3,5,7,9],[4,6,8,10]] [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]=>[1,2,3,4,6,5,8,7,10,9]=>[[1,2,3,4,5,7,9],[6,8,10]] [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]=>[1,2,3,4,5,6,8,7,10,9]=>[[1,2,3,4,5,6,7,9],[8,10]] [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]=>[1,2,3,4,5,6,7,8,10,9]=>[[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]=>[[1,2,3,4,5,6,7,8,9,10]] [12]=>[[1,2,3,4,5,6,7,8,9,10,11,12]]=>[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]] [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]=>[6,5,4,3,2,1,12,11,10,9,8,7]=>[[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]] [2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>[11,12,9,10,7,8,5,6,3,4,1,2]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[[1,3,5,7,9,11],[2,4,6,8,10,12]] [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]=>[[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
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
Robinson-Schensted recording tableau
Description
Sends a permutation to its Robinson-Schensted recording tableau.
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.
This map sends a permutation to its corresponding recording tableau.