Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ 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]=>[2,1]
[3]=>[[1,2,3]]=>[1,2,3]=>[1,2,3]
[2,1]=>[[1,2],[3]]=>[3,1,2]=>[3,1,2]
[1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>[2,3,1]
[4]=>[[1,2,3,4]]=>[1,2,3,4]=>[1,2,3,4]
[3,1]=>[[1,2,3],[4]]=>[4,1,2,3]=>[4,1,2,3]
[2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[4,1,3,2]
[2,1,1]=>[[1,2],[3],[4]]=>[4,3,1,2]=>[3,1,4,2]
[1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[2,3,4,1]
[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]=>[5,1,2,3,4]
[3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[5,1,2,4,3]
[3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[4,1,2,5,3]
[2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[4,1,5,3,2]
[2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[3,1,4,5,2]
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[2,3,4,5,1]
[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]=>[6,1,2,3,4,5]
[4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[6,1,2,3,5,4]
[4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[5,1,2,3,6,4]
[3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]
[3,2,1]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[5,1,2,6,4,3]
[3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[4,1,2,5,6,3]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[4,1,6,3,5,2]
[2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[4,1,5,3,6,2]
[2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[3,1,4,5,6,2]
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]
[7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7]
[6,1]=>[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[7,1,2,3,4,5,6]
[5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[7,1,2,3,4,6,5]
[5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[6,1,2,3,4,7,5]
[4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[7,1,2,3,5,6,4]
[4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[6,1,2,3,7,5,4]
[4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[5,1,2,3,6,7,4]
[3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[6,1,2,7,4,5,3]
[3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[5,1,2,7,4,6,3]
[3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[5,1,2,6,4,7,3]
[3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[4,1,2,5,6,7,3]
[2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[4,1,6,3,7,5,2]
[2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[4,1,5,3,6,7,2]
[2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[3,1,4,5,6,7,2]
[1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[2,3,4,5,6,7,1]
[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]=>[[1,2,3,4,5,6,7],[8]]=>[8,1,2,3,4,5,6,7]=>[8,1,2,3,4,5,6,7]
[6,2]=>[[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>[8,1,2,3,4,5,7,6]
[6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>[8,7,1,2,3,4,5,6]=>[7,1,2,3,4,5,8,6]
[5,3]=>[[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>[8,1,2,3,4,6,7,5]
[5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[7,1,2,3,4,8,6,5]
[5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[6,1,2,3,4,7,8,5]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[8,1,2,3,5,6,7,4]
[4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[7,1,2,3,8,5,6,4]
[4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[6,1,2,3,8,5,7,4]
[4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[6,1,2,3,7,5,8,4]
[4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[5,1,2,3,6,7,8,4]
[3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>[7,8,4,5,6,1,2,3]=>[6,1,2,8,4,5,7,3]
[3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>[8,7,4,5,6,1,2,3]=>[6,1,2,7,4,5,8,3]
[3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>[5,1,2,7,4,8,6,3]
[3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>[8,7,6,4,5,1,2,3]=>[5,1,2,6,4,7,8,3]
[3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>[4,1,2,5,6,7,8,3]
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,7,2]
[2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>[4,1,6,3,7,5,8,2]
[2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>[4,1,5,3,6,7,8,2]
[2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[3,1,4,5,6,7,8,2]
[1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[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]=>[[1,2,3,4,5,6,7,8],[9]]=>[9,1,2,3,4,5,6,7,8]=>[9,1,2,3,4,5,6,7,8]
[7,2]=>[[1,2,3,4,5,6,7],[8,9]]=>[8,9,1,2,3,4,5,6,7]=>[9,1,2,3,4,5,6,8,7]
[7,1,1]=>[[1,2,3,4,5,6,7],[8],[9]]=>[9,8,1,2,3,4,5,6,7]=>[8,1,2,3,4,5,6,9,7]
[6,3]=>[[1,2,3,4,5,6],[7,8,9]]=>[7,8,9,1,2,3,4,5,6]=>[9,1,2,3,4,5,7,8,6]
[6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>[9,7,8,1,2,3,4,5,6]=>[8,1,2,3,4,5,9,7,6]
[6,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9]]=>[9,8,7,1,2,3,4,5,6]=>[7,1,2,3,4,5,8,9,6]
[5,4]=>[[1,2,3,4,5],[6,7,8,9]]=>[6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,6,7,8,5]
[5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>[9,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,9,6,7,5]
[5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>[8,9,6,7,1,2,3,4,5]=>[7,1,2,3,4,9,6,8,5]
[5,2,1,1]=>[[1,2,3,4,5],[6,7],[8],[9]]=>[9,8,6,7,1,2,3,4,5]=>[7,1,2,3,4,8,6,9,5]
[5,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9]]=>[9,8,7,6,1,2,3,4,5]=>[6,1,2,3,4,7,8,9,5]
[4,4,1]=>[[1,2,3,4],[5,6,7,8],[9]]=>[9,5,6,7,8,1,2,3,4]=>[8,1,2,3,9,5,6,7,4]
[4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>[8,9,5,6,7,1,2,3,4]=>[7,1,2,3,9,5,6,8,4]
[4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>[9,8,5,6,7,1,2,3,4]=>[7,1,2,3,8,5,6,9,4]
[4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,1,2,3,4]=>[6,1,2,3,8,5,9,7,4]
[4,2,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,1,2,3,4]=>[6,1,2,3,7,5,8,9,4]
[4,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,1,2,3,4]=>[5,1,2,3,6,7,8,9,4]
[3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[7,8,9,4,5,6,1,2,3]=>[6,1,2,9,4,5,7,8,3]
[3,3,2,1]=>[[1,2,3],[4,5,6],[7,8],[9]]=>[9,7,8,4,5,6,1,2,3]=>[6,1,2,8,4,5,9,7,3]
[3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>[9,8,7,4,5,6,1,2,3]=>[6,1,2,7,4,5,8,9,3]
[3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>[8,9,6,7,4,5,1,2,3]=>[5,1,2,7,4,9,6,8,3]
[3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>[9,8,6,7,4,5,1,2,3]=>[5,1,2,7,4,8,6,9,3]
[3,2,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9]]=>[9,8,7,6,4,5,1,2,3]=>[5,1,2,6,4,7,8,9,3]
[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]=>[4,1,2,5,6,7,8,9,3]
[2,2,2,2,1]=>[[1,2],[3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,9,7,2]
[2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,3,4,1,2]=>[4,1,6,3,7,5,8,9,2]
[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]=>[4,1,5,3,6,7,8,9,2]
[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]=>[3,1,4,5,6,7,8,9,2]
[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]=>[2,3,4,5,6,7,8,9,1]
[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]=>[[1,2,3,4,5,6,7,8,9],[10]]=>[10,1,2,3,4,5,6,7,8,9]=>[10,1,2,3,4,5,6,7,8,9]
[8,2]=>[[1,2,3,4,5,6,7,8],[9,10]]=>[9,10,1,2,3,4,5,6,7,8]=>[10,1,2,3,4,5,6,7,9,8]
[8,1,1]=>[[1,2,3,4,5,6,7,8],[9],[10]]=>[10,9,1,2,3,4,5,6,7,8]=>[9,1,2,3,4,5,6,7,10,8]
[7,3]=>[[1,2,3,4,5,6,7],[8,9,10]]=>[8,9,10,1,2,3,4,5,6,7]=>[10,1,2,3,4,5,6,8,9,7]
[7,2,1]=>[[1,2,3,4,5,6,7],[8,9],[10]]=>[10,8,9,1,2,3,4,5,6,7]=>[9,1,2,3,4,5,6,10,8,7]
[7,1,1,1]=>[[1,2,3,4,5,6,7],[8],[9],[10]]=>[10,9,8,1,2,3,4,5,6,7]=>[8,1,2,3,4,5,6,9,10,7]
[6,4]=>[[1,2,3,4,5,6],[7,8,9,10]]=>[7,8,9,10,1,2,3,4,5,6]=>[10,1,2,3,4,5,7,8,9,6]
[6,3,1]=>[[1,2,3,4,5,6],[7,8,9],[10]]=>[10,7,8,9,1,2,3,4,5,6]=>[9,1,2,3,4,5,10,7,8,6]
[6,2,2]=>[[1,2,3,4,5,6],[7,8],[9,10]]=>[9,10,7,8,1,2,3,4,5,6]=>[8,1,2,3,4,5,10,7,9,6]
[6,2,1,1]=>[[1,2,3,4,5,6],[7,8],[9],[10]]=>[10,9,7,8,1,2,3,4,5,6]=>[8,1,2,3,4,5,9,7,10,6]
[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]=>[7,1,2,3,4,5,8,9,10,6]
[5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[6,7,8,9,10,1,2,3,4,5]=>[10,1,2,3,4,6,7,8,9,5]
[5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>[10,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,10,6,7,8,5]
[5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>[9,10,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,10,6,7,9,5]
[5,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10]]=>[10,9,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,9,6,7,10,5]
[5,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10]]=>[10,8,9,6,7,1,2,3,4,5]=>[7,1,2,3,4,9,6,10,8,5]
[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]=>[7,1,2,3,4,8,6,9,10,5]
[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]=>[6,1,2,3,4,7,8,9,10,5]
[4,4,2]=>[[1,2,3,4],[5,6,7,8],[9,10]]=>[9,10,5,6,7,8,1,2,3,4]=>[8,1,2,3,10,5,6,7,9,4]
[4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>[10,9,5,6,7,8,1,2,3,4]=>[8,1,2,3,9,5,6,7,10,4]
[4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>[8,9,10,5,6,7,1,2,3,4]=>[7,1,2,3,10,5,6,8,9,4]
[4,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10]]=>[10,8,9,5,6,7,1,2,3,4]=>[7,1,2,3,9,5,6,10,8,4]
[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]=>[7,1,2,3,8,5,6,9,10,4]
[4,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,1,2,3,4]=>[6,1,2,3,8,5,10,7,9,4]
[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]=>[6,1,2,3,8,5,9,7,10,4]
[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]=>[6,1,2,3,7,5,8,9,10,4]
[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]=>[5,1,2,3,6,7,8,9,10,4]
[3,3,3,1]=>[[1,2,3],[4,5,6],[7,8,9],[10]]=>[10,7,8,9,4,5,6,1,2,3]=>[6,1,2,9,4,5,10,7,8,3]
[3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>[9,10,7,8,4,5,6,1,2,3]=>[6,1,2,8,4,5,10,7,9,3]
[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]=>[6,1,2,8,4,5,9,7,10,3]
[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]=>[6,1,2,7,4,5,8,9,10,3]
[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]=>[5,1,2,7,4,9,6,10,8,3]
[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]=>[5,1,2,7,4,8,6,9,10,3]
[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]=>[5,1,2,6,4,7,8,9,10,3]
[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]=>[4,1,2,5,6,7,8,9,10,3]
[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]=>[4,1,6,3,8,5,10,7,9,2]
[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]=>[4,1,6,3,8,5,9,7,10,2]
[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]=>[4,1,6,3,7,5,8,9,10,2]
[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]=>[4,1,5,3,6,7,8,9,10,2]
[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]=>[3,1,4,5,6,7,8,9,10,2]
[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]=>[2,3,4,5,6,7,8,9,10,1]
[5,4,2]=>[[1,2,3,4,5],[6,7,8,9],[10,11]]=>[10,11,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,11,6,7,8,10,5]
[5,4,1,1]=>[[1,2,3,4,5],[6,7,8,9],[10],[11]]=>[11,10,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,10,6,7,8,11,5]
[5,3,3]=>[[1,2,3,4,5],[6,7,8],[9,10,11]]=>[9,10,11,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,11,6,7,9,10,5]
[5,3,2,1]=>[[1,2,3,4,5],[6,7,8],[9,10],[11]]=>[11,9,10,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,10,6,7,11,9,5]
[5,3,1,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10],[11]]=>[11,10,9,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,9,6,7,10,11,5]
[5,2,2,2]=>[[1,2,3,4,5],[6,7],[8,9],[10,11]]=>[10,11,8,9,6,7,1,2,3,4,5]=>[7,1,2,3,4,9,6,11,8,10,5]
[5,2,2,1,1]=>[[1,2,3,4,5],[6,7],[8,9],[10],[11]]=>[11,10,8,9,6,7,1,2,3,4,5]=>[7,1,2,3,4,9,6,10,8,11,5]
[4,4,3]=>[[1,2,3,4],[5,6,7,8],[9,10,11]]=>[9,10,11,5,6,7,8,1,2,3,4]=>[8,1,2,3,11,5,6,7,9,10,4]
[4,4,2,1]=>[[1,2,3,4],[5,6,7,8],[9,10],[11]]=>[11,9,10,5,6,7,8,1,2,3,4]=>[8,1,2,3,10,5,6,7,11,9,4]
[4,4,1,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10],[11]]=>[11,10,9,5,6,7,8,1,2,3,4]=>[8,1,2,3,9,5,6,7,10,11,4]
[4,3,3,1]=>[[1,2,3,4],[5,6,7],[8,9,10],[11]]=>[11,8,9,10,5,6,7,1,2,3,4]=>[7,1,2,3,10,5,6,11,8,9,4]
[4,3,2,2]=>[[1,2,3,4],[5,6,7],[8,9],[10,11]]=>[10,11,8,9,5,6,7,1,2,3,4]=>[7,1,2,3,9,5,6,11,8,10,4]
[4,3,2,1,1]=>[[1,2,3,4],[5,6,7],[8,9],[10],[11]]=>[11,10,8,9,5,6,7,1,2,3,4]=>[7,1,2,3,9,5,6,10,8,11,4]
[4,2,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9,10],[11]]=>[11,9,10,7,8,5,6,1,2,3,4]=>[6,1,2,3,8,5,10,7,11,9,4]
[3,3,3,2]=>[[1,2,3],[4,5,6],[7,8,9],[10,11]]=>[10,11,7,8,9,4,5,6,1,2,3]=>[6,1,2,9,4,5,11,7,8,10,3]
[3,3,3,1,1]=>[[1,2,3],[4,5,6],[7,8,9],[10],[11]]=>[11,10,7,8,9,4,5,6,1,2,3]=>[6,1,2,9,4,5,10,7,8,11,3]
[3,3,2,2,1]=>[[1,2,3],[4,5,6],[7,8],[9,10],[11]]=>[11,9,10,7,8,4,5,6,1,2,3]=>[6,1,2,8,4,5,10,7,11,9,3]
[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]=>[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]=>[12,1,2,3,4,5,7,8,9,10,11,6]
[6,4,2]=>[[1,2,3,4,5,6],[7,8,9,10],[11,12]]=>[11,12,7,8,9,10,1,2,3,4,5,6]=>[10,1,2,3,4,5,12,7,8,9,11,6]
[5,4,3]=>[[1,2,3,4,5],[6,7,8,9],[10,11,12]]=>[10,11,12,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,12,6,7,8,10,11,5]
[5,4,2,1]=>[[1,2,3,4,5],[6,7,8,9],[10,11],[12]]=>[12,10,11,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,11,6,7,8,12,10,5]
[5,4,1,1,1]=>[[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]=>[12,11,10,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,10,6,7,8,11,12,5]
[5,3,3,1]=>[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]=>[12,9,10,11,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,11,6,7,12,9,10,5]
[5,3,2,2]=>[[1,2,3,4,5],[6,7,8],[9,10],[11,12]]=>[11,12,9,10,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,10,6,7,12,9,11,5]
[5,3,2,1,1]=>[[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]=>[12,11,9,10,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,10,6,7,11,9,12,5]
[5,2,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]=>[12,10,11,8,9,6,7,1,2,3,4,5]=>[7,1,2,3,4,9,6,11,8,12,10,5]
[4,4,3,1]=>[[1,2,3,4],[5,6,7,8],[9,10,11],[12]]=>[12,9,10,11,5,6,7,8,1,2,3,4]=>[8,1,2,3,11,5,6,7,12,9,10,4]
[4,4,2,2]=>[[1,2,3,4],[5,6,7,8],[9,10],[11,12]]=>[11,12,9,10,5,6,7,8,1,2,3,4]=>[8,1,2,3,10,5,6,7,12,9,11,4]
[4,4,2,1,1]=>[[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]=>[12,11,9,10,5,6,7,8,1,2,3,4]=>[8,1,2,3,10,5,6,7,11,9,12,4]
[4,3,3,2]=>[[1,2,3,4],[5,6,7],[8,9,10],[11,12]]=>[11,12,8,9,10,5,6,7,1,2,3,4]=>[7,1,2,3,10,5,6,12,8,9,11,4]
[4,3,3,1,1]=>[[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]=>[12,11,8,9,10,5,6,7,1,2,3,4]=>[7,1,2,3,10,5,6,11,8,9,12,4]
[4,3,2,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]=>[12,10,11,8,9,5,6,7,1,2,3,4]=>[7,1,2,3,9,5,6,11,8,12,10,4]
[4,2,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]=>[11,12,9,10,7,8,5,6,1,2,3,4]=>[6,1,2,3,8,5,10,7,12,9,11,4]
[3,3,3,2,1]=>[[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]=>[12,10,11,7,8,9,4,5,6,1,2,3]=>[6,1,2,9,4,5,11,7,8,12,10,3]
[3,3,2,2,1,1]=>[[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]=>[12,11,9,10,7,8,4,5,6,1,2,3]=>[6,1,2,8,4,5,10,7,11,9,12,3]
[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]=>[4,1,6,3,8,5,10,7,12,9,11,2]
[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]=>[2,3,4,5,6,7,8,9,10,11,12,1]
[5,4,3,1]=>[[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]=>[13,10,11,12,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,12,6,7,8,13,10,11,5]
[5,4,2,2]=>[[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]=>[12,13,10,11,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,11,6,7,8,13,10,12,5]
[5,4,2,1,1]=>[[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]=>[13,12,10,11,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,11,6,7,8,12,10,13,5]
[5,3,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]=>[12,13,9,10,11,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,11,6,7,13,9,10,12,5]
[5,3,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]=>[13,12,9,10,11,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,11,6,7,12,9,10,13,5]
[5,3,2,2,1]=>[[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]=>[13,11,12,9,10,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,10,6,7,12,9,13,11,5]
[4,4,3,2]=>[[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]=>[12,13,9,10,11,5,6,7,8,1,2,3,4]=>[8,1,2,3,11,5,6,7,13,9,10,12,4]
[4,4,3,1,1]=>[[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]=>[13,12,9,10,11,5,6,7,8,1,2,3,4]=>[8,1,2,3,11,5,6,7,12,9,10,13,4]
[4,4,2,2,1]=>[[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]=>[13,11,12,9,10,5,6,7,8,1,2,3,4]=>[8,1,2,3,10,5,6,7,12,9,13,11,4]
[4,3,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]=>[13,11,12,8,9,10,5,6,7,1,2,3,4]=>[7,1,2,3,10,5,6,12,8,9,13,11,4]
[7,7]=>[[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]=>[8,9,10,11,12,13,14,1,2,3,4,5,6,7]=>[14,1,2,3,4,5,6,8,9,10,11,12,13,7]
[5,4,3,2]=>[[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]=>[13,14,10,11,12,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,12,6,7,8,14,10,11,13,5]
[5,4,3,1,1]=>[[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]=>[14,13,10,11,12,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,12,6,7,8,13,10,11,14,5]
[5,4,2,2,1]=>[[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]=>[14,12,13,10,11,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,11,6,7,8,13,10,14,12,5]
[5,3,3,2,1]=>[[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]=>[14,12,13,9,10,11,6,7,8,1,2,3,4,5]=>[8,1,2,3,4,11,6,7,13,9,10,14,12,5]
[4,4,3,2,1]=>[[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]=>[14,12,13,9,10,11,5,6,7,8,1,2,3,4]=>[8,1,2,3,11,5,6,7,13,9,10,14,12,4]
[5,4,3,2,1]=>[[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]=>[15,13,14,10,11,12,6,7,8,9,1,2,3,4,5]=>[9,1,2,3,4,12,6,7,8,14,10,11,15,13,5]
[]=>[]=>[]=>[]
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
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
- the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
- the set of left-to-right maxima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
- the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
- the set of maximal elements in the decreasing runs of $\pi$ is the set of weak deficiency positions of $\chi(\pi)$, and
- the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.
searching the database
Sorry, this map was not found in the database.