Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[1]=>[[1]]=>[1]=>[1]
[2]=>[[1,2]]=>[1,2]=>[2]
[1,1]=>[[1],[2]]=>[2,1]=>[1,1]
[3]=>[[1,2,3]]=>[1,2,3]=>[3]
[2,1]=>[[1,2],[3]]=>[3,1,2]=>[1,2]
[1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>[1,1,1]
[4]=>[[1,2,3,4]]=>[1,2,3,4]=>[4]
[3,1]=>[[1,2,3],[4]]=>[4,1,2,3]=>[1,3]
[2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,2]
[2,1,1]=>[[1,2],[3],[4]]=>[4,3,1,2]=>[1,1,2]
[1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[1,1,1,1]
[5]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>[5]
[4,1]=>[[1,2,3,4],[5]]=>[5,1,2,3,4]=>[1,4]
[3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[2,3]
[3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[1,1,3]
[2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[1,2,2]
[2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[1,1,1,2]
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[1,1,1,1,1]
[6]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[6]
[5,1]=>[[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[1,5]
[4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[2,4]
[4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[1,1,4]
[3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[3,3]
[3,2,1]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[1,2,3]
[3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[1,1,1,3]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[2,2,2]
[2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[1,1,2,2]
[2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[1,1,1,1,2]
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[1,1,1,1,1,1]
[7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[7]
[6,1]=>[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[1,6]
[5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[2,5]
[5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[1,1,5]
[4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[3,4]
[4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[1,2,4]
[4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[1,1,1,4]
[3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[1,3,3]
[3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[2,2,3]
[3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[1,1,2,3]
[3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[1,1,1,1,3]
[2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[1,2,2,2]
[2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[1,1,1,2,2]
[2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[1,1,1,1,1,2]
[1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[1,1,1,1,1,1,1]
[8]=>[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[8]
[7,1]=>[[1,2,3,4,5,6,7],[8]]=>[8,1,2,3,4,5,6,7]=>[1,7]
[6,2]=>[[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>[2,6]
[6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>[8,7,1,2,3,4,5,6]=>[1,1,6]
[5,3]=>[[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>[3,5]
[5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[1,2,5]
[5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[1,1,1,5]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[4,4]
[4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[1,3,4]
[4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[2,2,4]
[4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[1,1,2,4]
[4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[1,1,1,1,4]
[3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>[7,8,4,5,6,1,2,3]=>[2,3,3]
[3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>[8,7,4,5,6,1,2,3]=>[1,1,3,3]
[3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>[1,2,2,3]
[3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>[8,7,6,4,5,1,2,3]=>[1,1,1,2,3]
[3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>[1,1,1,1,1,3]
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[2,2,2,2]
[2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>[1,1,2,2,2]
[2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>[1,1,1,1,2,2]
[2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[1,1,1,1,1,1,2]
[1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[1,1,1,1,1,1,1,1]
[9]=>[[1,2,3,4,5,6,7,8,9]]=>[1,2,3,4,5,6,7,8,9]=>[9]
[8,1]=>[[1,2,3,4,5,6,7,8],[9]]=>[9,1,2,3,4,5,6,7,8]=>[1,8]
[7,2]=>[[1,2,3,4,5,6,7],[8,9]]=>[8,9,1,2,3,4,5,6,7]=>[2,7]
[7,1,1]=>[[1,2,3,4,5,6,7],[8],[9]]=>[9,8,1,2,3,4,5,6,7]=>[1,1,7]
[6,3]=>[[1,2,3,4,5,6],[7,8,9]]=>[7,8,9,1,2,3,4,5,6]=>[3,6]
[6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>[9,7,8,1,2,3,4,5,6]=>[1,2,6]
[6,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9]]=>[9,8,7,1,2,3,4,5,6]=>[1,1,1,6]
[5,4]=>[[1,2,3,4,5],[6,7,8,9]]=>[6,7,8,9,1,2,3,4,5]=>[4,5]
[5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>[9,6,7,8,1,2,3,4,5]=>[1,3,5]
[5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>[8,9,6,7,1,2,3,4,5]=>[2,2,5]
[5,2,1,1]=>[[1,2,3,4,5],[6,7],[8],[9]]=>[9,8,6,7,1,2,3,4,5]=>[1,1,2,5]
[5,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9]]=>[9,8,7,6,1,2,3,4,5]=>[1,1,1,1,5]
[4,4,1]=>[[1,2,3,4],[5,6,7,8],[9]]=>[9,5,6,7,8,1,2,3,4]=>[1,4,4]
[4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>[8,9,5,6,7,1,2,3,4]=>[2,3,4]
[4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>[9,8,5,6,7,1,2,3,4]=>[1,1,3,4]
[4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,1,2,3,4]=>[1,2,2,4]
[4,2,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,1,2,3,4]=>[1,1,1,2,4]
[4,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,1,2,3,4]=>[1,1,1,1,1,4]
[3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[7,8,9,4,5,6,1,2,3]=>[3,3,3]
[3,3,2,1]=>[[1,2,3],[4,5,6],[7,8],[9]]=>[9,7,8,4,5,6,1,2,3]=>[1,2,3,3]
[3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>[9,8,7,4,5,6,1,2,3]=>[1,1,1,3,3]
[3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>[8,9,6,7,4,5,1,2,3]=>[2,2,2,3]
[3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>[9,8,6,7,4,5,1,2,3]=>[1,1,2,2,3]
[3,2,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9]]=>[9,8,7,6,4,5,1,2,3]=>[1,1,1,1,2,3]
[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]=>[1,1,1,1,1,1,3]
[2,2,2,2,1]=>[[1,2],[3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,3,4,1,2]=>[1,2,2,2,2]
[2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,3,4,1,2]=>[1,1,1,2,2,2]
[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]=>[1,1,1,1,1,2,2]
[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]=>[1,1,1,1,1,1,1,2]
[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]=>[1,1,1,1,1,1,1,1,1]
[10]=>[[1,2,3,4,5,6,7,8,9,10]]=>[1,2,3,4,5,6,7,8,9,10]=>[10]
[9,1]=>[[1,2,3,4,5,6,7,8,9],[10]]=>[10,1,2,3,4,5,6,7,8,9]=>[1,9]
[8,2]=>[[1,2,3,4,5,6,7,8],[9,10]]=>[9,10,1,2,3,4,5,6,7,8]=>[2,8]
[8,1,1]=>[[1,2,3,4,5,6,7,8],[9],[10]]=>[10,9,1,2,3,4,5,6,7,8]=>[1,1,8]
[7,3]=>[[1,2,3,4,5,6,7],[8,9,10]]=>[8,9,10,1,2,3,4,5,6,7]=>[3,7]
[7,2,1]=>[[1,2,3,4,5,6,7],[8,9],[10]]=>[10,8,9,1,2,3,4,5,6,7]=>[1,2,7]
[7,1,1,1]=>[[1,2,3,4,5,6,7],[8],[9],[10]]=>[10,9,8,1,2,3,4,5,6,7]=>[1,1,1,7]
[6,4]=>[[1,2,3,4,5,6],[7,8,9,10]]=>[7,8,9,10,1,2,3,4,5,6]=>[4,6]
[6,3,1]=>[[1,2,3,4,5,6],[7,8,9],[10]]=>[10,7,8,9,1,2,3,4,5,6]=>[1,3,6]
[6,2,2]=>[[1,2,3,4,5,6],[7,8],[9,10]]=>[9,10,7,8,1,2,3,4,5,6]=>[2,2,6]
[6,2,1,1]=>[[1,2,3,4,5,6],[7,8],[9],[10]]=>[10,9,7,8,1,2,3,4,5,6]=>[1,1,2,6]
[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]=>[1,1,1,1,6]
[5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[6,7,8,9,10,1,2,3,4,5]=>[5,5]
[5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>[10,6,7,8,9,1,2,3,4,5]=>[1,4,5]
[5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>[9,10,6,7,8,1,2,3,4,5]=>[2,3,5]
[5,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10]]=>[10,9,6,7,8,1,2,3,4,5]=>[1,1,3,5]
[5,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10]]=>[10,8,9,6,7,1,2,3,4,5]=>[1,2,2,5]
[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]=>[1,1,1,2,5]
[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]=>[1,1,1,1,1,5]
[4,4,2]=>[[1,2,3,4],[5,6,7,8],[9,10]]=>[9,10,5,6,7,8,1,2,3,4]=>[2,4,4]
[4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>[10,9,5,6,7,8,1,2,3,4]=>[1,1,4,4]
[4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>[8,9,10,5,6,7,1,2,3,4]=>[3,3,4]
[4,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10]]=>[10,8,9,5,6,7,1,2,3,4]=>[1,2,3,4]
[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]=>[1,1,1,3,4]
[4,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,1,2,3,4]=>[2,2,2,4]
[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]=>[1,1,2,2,4]
[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]=>[1,1,1,1,2,4]
[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]=>[1,1,1,1,1,1,4]
[3,3,3,1]=>[[1,2,3],[4,5,6],[7,8,9],[10]]=>[10,7,8,9,4,5,6,1,2,3]=>[1,3,3,3]
[3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>[9,10,7,8,4,5,6,1,2,3]=>[2,2,3,3]
[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]=>[1,1,2,3,3]
[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]=>[1,1,1,1,3,3]
[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]=>[1,2,2,2,3]
[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]=>[1,1,1,2,2,3]
[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]=>[1,1,1,1,1,2,3]
[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]=>[1,1,1,1,1,1,1,3]
[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]=>[2,2,2,2,2]
[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]=>[1,1,2,2,2,2]
[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]=>[1,1,1,1,2,2,2]
[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]=>[1,1,1,1,1,1,2,2]
[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]=>[1,1,1,1,1,1,1,1,2]
[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]=>[1,1,1,1,1,1,1,1,1,1]
[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]=>[2,4,5]
[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]=>[1,1,4,5]
[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]=>[3,3,5]
[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]=>[1,2,3,5]
[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]=>[1,1,1,3,5]
[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]=>[2,2,2,5]
[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]=>[1,1,2,2,5]
[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]=>[3,4,4]
[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]=>[1,2,4,4]
[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]=>[1,1,1,4,4]
[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]=>[1,3,3,4]
[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]=>[2,2,3,4]
[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]=>[1,1,2,3,4]
[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]=>[1,2,2,2,4]
[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]=>[2,3,3,3]
[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]=>[1,1,3,3,3]
[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]=>[1,2,2,3,3]
[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]=>[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]=>[6,6]
[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]=>[2,4,6]
[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]=>[3,4,5]
[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]=>[1,2,4,5]
[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]=>[1,1,1,4,5]
[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]=>[1,3,3,5]
[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]=>[2,2,3,5]
[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]=>[1,1,2,3,5]
[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]=>[1,2,2,2,5]
[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]=>[1,3,4,4]
[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]=>[2,2,4,4]
[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]=>[1,1,2,4,4]
[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]=>[2,3,3,4]
[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]=>[1,1,3,3,4]
[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]=>[1,2,2,3,4]
[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]=>[2,2,2,2,4]
[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]=>[1,2,3,3,3]
[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]=>[1,1,2,2,3,3]
[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]=>[2,2,2,2,2,2]
[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]=>[1,1,1,1,1,1,1,1,1,1,1,1]
[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]=>[1,3,4,5]
[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]=>[2,2,4,5]
[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]=>[1,1,2,4,5]
[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]=>[2,3,3,5]
[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]=>[1,1,3,3,5]
[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]=>[1,2,2,3,5]
[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]=>[2,3,4,4]
[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]=>[1,1,3,4,4]
[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]=>[1,2,2,4,4]
[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]=>[1,2,3,3,4]
[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]=>[2,3,4,5]
[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]=>[1,1,3,4,5]
[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]=>[1,2,2,4,5]
[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]=>[1,2,3,3,5]
[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]=>[1,2,3,4,4]
[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]=>[1,2,3,4,5]
[]=>[]=>[]=>[]
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 composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
searching the database
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