Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1Cc0002;cc-rep-3
[1]=>[[1]]=>[1]=>[1]
[2]=>[[1,2]]=>[1,2]=>[1,1]
[1,1]=>[[1],[2]]=>[2,1]=>[2]
[3]=>[[1,2,3]]=>[1,2,3]=>[1,1,1]
[2,1]=>[[1,2],[3]]=>[3,1,2]=>[2,1]
[1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>[3]
[4]=>[[1,2,3,4]]=>[1,2,3,4]=>[1,1,1,1]
[3,1]=>[[1,2,3],[4]]=>[4,1,2,3]=>[2,1,1]
[2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,1,1]
[2,1,1]=>[[1,2],[3],[4]]=>[4,3,1,2]=>[3,1]
[1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[4]
[5]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>[1,1,1,1,1]
[4,1]=>[[1,2,3,4],[5]]=>[5,1,2,3,4]=>[2,1,1,1]
[3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[2,1,1,1]
[3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[3,1,1]
[2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[3,1,1]
[2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[4,1]
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[5]
[6]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[1,1,1,1,1,1]
[5,1]=>[[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[2,1,1,1,1]
[4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[2,1,1,1,1]
[4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[3,1,1,1]
[3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[2,1,1,1,1]
[3,2,1]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[3,1,1,1]
[3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[4,1,1]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[3,1,1,1]
[2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[4,1,1]
[2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[5,1]
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[6]
[7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[1,1,1,1,1,1,1]
[6,1]=>[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[2,1,1,1,1,1]
[5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[2,1,1,1,1,1]
[5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[3,1,1,1,1]
[4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[2,1,1,1,1,1]
[4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[3,1,1,1,1]
[4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[4,1,1,1]
[3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[3,1,1,1,1]
[3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[3,1,1,1,1]
[3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[4,1,1,1]
[3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[5,1,1]
[2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[4,1,1,1]
[2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[5,1,1]
[2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[6,1]
[1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[7]
[8]=>[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[1,1,1,1,1,1,1,1]
[7,1]=>[[1,2,3,4,5,6,7],[8]]=>[8,1,2,3,4,5,6,7]=>[2,1,1,1,1,1,1]
[6,2]=>[[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>[2,1,1,1,1,1,1]
[6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>[8,7,1,2,3,4,5,6]=>[3,1,1,1,1,1]
[5,3]=>[[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>[2,1,1,1,1,1,1]
[5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[3,1,1,1,1,1]
[5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[4,1,1,1,1]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[2,1,1,1,1,1,1]
[4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[3,1,1,1,1,1]
[4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[3,1,1,1,1,1]
[4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[4,1,1,1,1]
[4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[5,1,1,1]
[3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>[7,8,4,5,6,1,2,3]=>[3,1,1,1,1,1]
[3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>[8,7,4,5,6,1,2,3]=>[4,1,1,1,1]
[3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>[4,1,1,1,1]
[3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>[8,7,6,4,5,1,2,3]=>[5,1,1,1]
[3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>[6,1,1]
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[4,1,1,1,1]
[2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>[5,1,1,1]
[2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>[6,1,1]
[2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[7,1]
[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]
[9]=>[[1,2,3,4,5,6,7,8,9]]=>[1,2,3,4,5,6,7,8,9]=>[1,1,1,1,1,1,1,1,1]
[8,1]=>[[1,2,3,4,5,6,7,8],[9]]=>[9,1,2,3,4,5,6,7,8]=>[2,1,1,1,1,1,1,1]
[7,2]=>[[1,2,3,4,5,6,7],[8,9]]=>[8,9,1,2,3,4,5,6,7]=>[2,1,1,1,1,1,1,1]
[7,1,1]=>[[1,2,3,4,5,6,7],[8],[9]]=>[9,8,1,2,3,4,5,6,7]=>[3,1,1,1,1,1,1]
[6,3]=>[[1,2,3,4,5,6],[7,8,9]]=>[7,8,9,1,2,3,4,5,6]=>[2,1,1,1,1,1,1,1]
[6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>[9,7,8,1,2,3,4,5,6]=>[3,1,1,1,1,1,1]
[6,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9]]=>[9,8,7,1,2,3,4,5,6]=>[4,1,1,1,1,1]
[5,4]=>[[1,2,3,4,5],[6,7,8,9]]=>[6,7,8,9,1,2,3,4,5]=>[2,1,1,1,1,1,1,1]
[5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>[9,6,7,8,1,2,3,4,5]=>[3,1,1,1,1,1,1]
[5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>[8,9,6,7,1,2,3,4,5]=>[3,1,1,1,1,1,1]
[5,2,1,1]=>[[1,2,3,4,5],[6,7],[8],[9]]=>[9,8,6,7,1,2,3,4,5]=>[4,1,1,1,1,1]
[5,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9]]=>[9,8,7,6,1,2,3,4,5]=>[5,1,1,1,1]
[4,4,1]=>[[1,2,3,4],[5,6,7,8],[9]]=>[9,5,6,7,8,1,2,3,4]=>[3,1,1,1,1,1,1]
[4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>[8,9,5,6,7,1,2,3,4]=>[3,1,1,1,1,1,1]
[4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>[9,8,5,6,7,1,2,3,4]=>[4,1,1,1,1,1]
[4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,1,2,3,4]=>[4,1,1,1,1,1]
[4,2,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,1,2,3,4]=>[5,1,1,1,1]
[4,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,1,2,3,4]=>[6,1,1,1]
[3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[7,8,9,4,5,6,1,2,3]=>[3,1,1,1,1,1,1]
[3,3,2,1]=>[[1,2,3],[4,5,6],[7,8],[9]]=>[9,7,8,4,5,6,1,2,3]=>[4,1,1,1,1,1]
[3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>[9,8,7,4,5,6,1,2,3]=>[5,1,1,1,1]
[3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>[8,9,6,7,4,5,1,2,3]=>[4,1,1,1,1,1]
[3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>[9,8,6,7,4,5,1,2,3]=>[5,1,1,1,1]
[3,2,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9]]=>[9,8,7,6,4,5,1,2,3]=>[6,1,1,1]
[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]=>[7,1,1]
[2,2,2,2,1]=>[[1,2],[3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,3,4,1,2]=>[5,1,1,1,1]
[2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,3,4,1,2]=>[6,1,1,1]
[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]=>[7,1,1]
[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]=>[8,1]
[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]
[10]=>[[1,2,3,4,5,6,7,8,9,10]]=>[1,2,3,4,5,6,7,8,9,10]=>[1,1,1,1,1,1,1,1,1,1]
[9,1]=>[[1,2,3,4,5,6,7,8,9],[10]]=>[10,1,2,3,4,5,6,7,8,9]=>[2,1,1,1,1,1,1,1,1]
[8,2]=>[[1,2,3,4,5,6,7,8],[9,10]]=>[9,10,1,2,3,4,5,6,7,8]=>[2,1,1,1,1,1,1,1,1]
[8,1,1]=>[[1,2,3,4,5,6,7,8],[9],[10]]=>[10,9,1,2,3,4,5,6,7,8]=>[3,1,1,1,1,1,1,1]
[7,3]=>[[1,2,3,4,5,6,7],[8,9,10]]=>[8,9,10,1,2,3,4,5,6,7]=>[2,1,1,1,1,1,1,1,1]
[7,2,1]=>[[1,2,3,4,5,6,7],[8,9],[10]]=>[10,8,9,1,2,3,4,5,6,7]=>[3,1,1,1,1,1,1,1]
[7,1,1,1]=>[[1,2,3,4,5,6,7],[8],[9],[10]]=>[10,9,8,1,2,3,4,5,6,7]=>[4,1,1,1,1,1,1]
[6,4]=>[[1,2,3,4,5,6],[7,8,9,10]]=>[7,8,9,10,1,2,3,4,5,6]=>[2,1,1,1,1,1,1,1,1]
[6,3,1]=>[[1,2,3,4,5,6],[7,8,9],[10]]=>[10,7,8,9,1,2,3,4,5,6]=>[3,1,1,1,1,1,1,1]
[6,2,2]=>[[1,2,3,4,5,6],[7,8],[9,10]]=>[9,10,7,8,1,2,3,4,5,6]=>[3,1,1,1,1,1,1,1]
[6,2,1,1]=>[[1,2,3,4,5,6],[7,8],[9],[10]]=>[10,9,7,8,1,2,3,4,5,6]=>[4,1,1,1,1,1,1]
[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]=>[5,1,1,1,1,1]
[5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[6,7,8,9,10,1,2,3,4,5]=>[2,1,1,1,1,1,1,1,1]
[5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>[10,6,7,8,9,1,2,3,4,5]=>[3,1,1,1,1,1,1,1]
[5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>[9,10,6,7,8,1,2,3,4,5]=>[3,1,1,1,1,1,1,1]
[5,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10]]=>[10,9,6,7,8,1,2,3,4,5]=>[4,1,1,1,1,1,1]
[5,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10]]=>[10,8,9,6,7,1,2,3,4,5]=>[4,1,1,1,1,1,1]
[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]=>[5,1,1,1,1,1]
[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]=>[6,1,1,1,1]
[4,4,2]=>[[1,2,3,4],[5,6,7,8],[9,10]]=>[9,10,5,6,7,8,1,2,3,4]=>[3,1,1,1,1,1,1,1]
[4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>[10,9,5,6,7,8,1,2,3,4]=>[4,1,1,1,1,1,1]
[4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>[8,9,10,5,6,7,1,2,3,4]=>[3,1,1,1,1,1,1,1]
[4,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10]]=>[10,8,9,5,6,7,1,2,3,4]=>[4,1,1,1,1,1,1]
[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]=>[5,1,1,1,1,1]
[4,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,1,2,3,4]=>[4,1,1,1,1,1,1]
[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]=>[5,1,1,1,1,1]
[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]=>[6,1,1,1,1]
[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]=>[7,1,1,1]
[3,3,3,1]=>[[1,2,3],[4,5,6],[7,8,9],[10]]=>[10,7,8,9,4,5,6,1,2,3]=>[4,1,1,1,1,1,1]
[3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>[9,10,7,8,4,5,6,1,2,3]=>[4,1,1,1,1,1,1]
[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]=>[5,1,1,1,1,1]
[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]=>[6,1,1,1,1]
[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]=>[5,1,1,1,1,1]
[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]=>[6,1,1,1,1]
[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]=>[7,1,1,1]
[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]=>[8,1,1]
[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]=>[5,1,1,1,1,1]
[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]=>[6,1,1,1,1]
[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]=>[7,1,1,1]
[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]=>[8,1,1]
[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]=>[9,1]
[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]
[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]=>[1,1,1,1,1,1,1,1,1,1,1,1]
[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]=>[2,1,1,1,1,1,1,1,1,1,1]
[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]=>[3,1,1,1,1,1,1,1,1,1]
[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]=>[4,1,1,1,1,1,1,1,1]
[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]=>[5,1,1,1,1,1,1,1]
[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]=>[6,1,1,1,1,1,1]
[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]=>[12]
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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
LLPS
Description
The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
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