Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1Cc0002;cc-rep-4
[1]=>[[1]]=>[1]=>[1]=>[1]
[2]=>[[1,2]]=>[1,2]=>[1,2]=>[1,1]
[1,1]=>[[1],[2]]=>[2,1]=>[2,1]=>[2]
[3]=>[[1,2,3]]=>[1,2,3]=>[1,2,3]=>[1,1,1]
[2,1]=>[[1,2],[3]]=>[3,1,2]=>[1,3,2]=>[2,1]
[1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>[3,2,1]=>[3]
[4]=>[[1,2,3,4]]=>[1,2,3,4]=>[1,2,3,4]=>[1,1,1,1]
[3,1]=>[[1,2,3],[4]]=>[4,1,2,3]=>[1,2,4,3]=>[2,1,1]
[2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[3,1,4,2]=>[2,2]
[2,1,1]=>[[1,2],[3],[4]]=>[4,3,1,2]=>[1,4,3,2]=>[3,1]
[1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[4,3,2,1]=>[4]
[5]=>[[1,2,3,4,5]]=>[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]=>[1,2,3,5,4]=>[2,1,1,1]
[3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[1,4,2,5,3]=>[2,2,1]
[3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[1,2,5,4,3]=>[3,1,1]
[2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[3,1,5,4,2]=>[3,2]
[2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[1,5,4,3,2]=>[4,1]
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[5,4,3,2,1]=>[5]
[6]=>[[1,2,3,4,5,6]]=>[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]=>[1,2,3,4,6,5]=>[2,1,1,1,1]
[4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[1,2,5,3,6,4]=>[2,2,1,1]
[4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[1,2,3,6,5,4]=>[3,1,1,1]
[3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>[2,2,2]
[3,2,1]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[1,4,2,6,5,3]=>[3,2,1]
[3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[1,2,6,5,4,3]=>[4,1,1]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[5,3,1,6,4,2]=>[3,3]
[2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[3,1,6,5,4,2]=>[4,2]
[2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[1,6,5,4,3,2]=>[5,1]
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[6]
[7]=>[[1,2,3,4,5,6,7]]=>[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]=>[1,2,3,4,5,7,6]=>[2,1,1,1,1,1]
[5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[1,2,3,6,4,7,5]=>[2,2,1,1,1]
[5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[1,2,3,4,7,6,5]=>[3,1,1,1,1]
[4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[1,5,2,6,3,7,4]=>[2,2,2,1]
[4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[1,2,5,3,7,6,4]=>[3,2,1,1]
[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]=>[4,1,1,1]
[3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[4,1,5,2,7,6,3]=>[3,2,2]
[3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[1,6,4,2,7,5,3]=>[3,3,1]
[3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[1,4,2,7,6,5,3]=>[4,2,1]
[3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[1,2,7,6,5,4,3]=>[5,1,1]
[2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[5,3,1,7,6,4,2]=>[4,3]
[2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[3,1,7,6,5,4,2]=>[5,2]
[2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[1,7,6,5,4,3,2]=>[6,1]
[1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[7,6,5,4,3,2,1]=>[7]
[8]=>[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8]=>[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]=>[1,2,3,4,5,6,8,7]=>[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]=>[1,2,3,4,7,5,8,6]=>[2,2,1,1,1,1]
[6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>[8,7,1,2,3,4,5,6]=>[1,2,3,4,5,8,7,6]=>[3,1,1,1,1,1]
[5,3]=>[[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>[1,2,6,3,7,4,8,5]=>[2,2,2,1,1]
[5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[1,2,3,6,4,8,7,5]=>[3,2,1,1,1]
[5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[1,2,3,4,8,7,6,5]=>[4,1,1,1,1]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>[2,2,2,2]
[4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[1,5,2,6,3,8,7,4]=>[3,2,2,1]
[4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[1,2,7,5,3,8,6,4]=>[3,3,1,1]
[4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[1,2,5,3,8,7,6,4]=>[4,2,1,1]
[4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[1,2,3,8,7,6,5,4]=>[5,1,1,1]
[3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>[7,8,4,5,6,1,2,3]=>[4,1,7,5,2,8,6,3]=>[3,3,2]
[3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>[8,7,4,5,6,1,2,3]=>[4,1,5,2,8,7,6,3]=>[4,2,2]
[3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>[1,6,4,2,8,7,5,3]=>[4,3,1]
[3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>[8,7,6,4,5,1,2,3]=>[1,4,2,8,7,6,5,3]=>[5,2,1]
[3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>[1,2,8,7,6,5,4,3]=>[6,1,1]
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[7,5,3,1,8,6,4,2]=>[4,4]
[2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>[5,3,1,8,7,6,4,2]=>[5,3]
[2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>[3,1,8,7,6,5,4,2]=>[6,2]
[2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[1,8,7,6,5,4,3,2]=>[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]=>[8,7,6,5,4,3,2,1]=>[8]
[9]=>[[1,2,3,4,5,6,7,8,9]]=>[1,2,3,4,5,6,7,8,9]=>[1,2,3,4,5,6,7,8,9]=>[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]=>[1,2,3,4,5,6,7,9,8]=>[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]=>[1,2,3,4,5,8,6,9,7]=>[2,2,1,1,1,1,1]
[7,1,1]=>[[1,2,3,4,5,6,7],[8],[9]]=>[9,8,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,9,8,7]=>[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]=>[1,2,3,7,4,8,5,9,6]=>[2,2,2,1,1,1]
[6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>[9,7,8,1,2,3,4,5,6]=>[1,2,3,4,7,5,9,8,6]=>[3,2,1,1,1,1]
[6,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9]]=>[9,8,7,1,2,3,4,5,6]=>[1,2,3,4,5,9,8,7,6]=>[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]=>[1,6,2,7,3,8,4,9,5]=>[2,2,2,2,1]
[5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>[9,6,7,8,1,2,3,4,5]=>[1,2,6,3,7,4,9,8,5]=>[3,2,2,1,1]
[5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>[8,9,6,7,1,2,3,4,5]=>[1,2,3,8,6,4,9,7,5]=>[3,3,1,1,1]
[5,2,1,1]=>[[1,2,3,4,5],[6,7],[8],[9]]=>[9,8,6,7,1,2,3,4,5]=>[1,2,3,6,4,9,8,7,5]=>[4,2,1,1,1]
[5,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9]]=>[9,8,7,6,1,2,3,4,5]=>[1,2,3,4,9,8,7,6,5]=>[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]=>[5,1,6,2,7,3,9,8,4]=>[3,2,2,2]
[4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>[8,9,5,6,7,1,2,3,4]=>[1,5,2,8,6,3,9,7,4]=>[3,3,2,1]
[4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>[9,8,5,6,7,1,2,3,4]=>[1,5,2,6,3,9,8,7,4]=>[4,2,2,1]
[4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,1,2,3,4]=>[1,2,7,5,3,9,8,6,4]=>[4,3,1,1]
[4,2,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,1,2,3,4]=>[1,2,5,3,9,8,7,6,4]=>[5,2,1,1]
[4,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,1,2,3,4]=>[1,2,3,9,8,7,6,5,4]=>[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]=>[7,4,1,8,5,2,9,6,3]=>[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]=>[4,1,7,5,2,9,8,6,3]=>[4,3,2]
[3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>[9,8,7,4,5,6,1,2,3]=>[4,1,5,2,9,8,7,6,3]=>[5,2,2]
[3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>[8,9,6,7,4,5,1,2,3]=>[1,8,6,4,2,9,7,5,3]=>[4,4,1]
[3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>[9,8,6,7,4,5,1,2,3]=>[1,6,4,2,9,8,7,5,3]=>[5,3,1]
[3,2,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9]]=>[9,8,7,6,4,5,1,2,3]=>[1,4,2,9,8,7,6,5,3]=>[6,2,1]
[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]=>[1,2,9,8,7,6,5,4,3]=>[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]=>[7,5,3,1,9,8,6,4,2]=>[5,4]
[2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,3,4,1,2]=>[5,3,1,9,8,7,6,4,2]=>[6,3]
[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]=>[3,1,9,8,7,6,5,4,2]=>[7,2]
[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]=>[1,9,8,7,6,5,4,3,2]=>[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]=>[9,8,7,6,5,4,3,2,1]=>[9]
[10]=>[[1,2,3,4,5,6,7,8,9,10]]=>[1,2,3,4,5,6,7,8,9,10]=>[1,2,3,4,5,6,7,8,9,10]=>[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]=>[1,2,3,4,5,6,7,8,10,9]=>[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]=>[1,2,3,4,5,6,9,7,10,8]=>[2,2,1,1,1,1,1,1]
[8,1,1]=>[[1,2,3,4,5,6,7,8],[9],[10]]=>[10,9,1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,10,9,8]=>[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]=>[1,2,3,4,8,5,9,6,10,7]=>[2,2,2,1,1,1,1]
[7,2,1]=>[[1,2,3,4,5,6,7],[8,9],[10]]=>[10,8,9,1,2,3,4,5,6,7]=>[1,2,3,4,5,8,6,10,9,7]=>[3,2,1,1,1,1,1]
[7,1,1,1]=>[[1,2,3,4,5,6,7],[8],[9],[10]]=>[10,9,8,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,10,9,8,7]=>[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]=>[1,2,7,3,8,4,9,5,10,6]=>[2,2,2,2,1,1]
[6,3,1]=>[[1,2,3,4,5,6],[7,8,9],[10]]=>[10,7,8,9,1,2,3,4,5,6]=>[1,2,3,7,4,8,5,10,9,6]=>[3,2,2,1,1,1]
[6,2,2]=>[[1,2,3,4,5,6],[7,8],[9,10]]=>[9,10,7,8,1,2,3,4,5,6]=>[1,2,3,4,9,7,5,10,8,6]=>[3,3,1,1,1,1]
[6,2,1,1]=>[[1,2,3,4,5,6],[7,8],[9],[10]]=>[10,9,7,8,1,2,3,4,5,6]=>[1,2,3,4,7,5,10,9,8,6]=>[4,2,1,1,1,1]
[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]=>[1,2,3,4,5,10,9,8,7,6]=>[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]=>[6,1,7,2,8,3,9,4,10,5]=>[2,2,2,2,2]
[5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>[10,6,7,8,9,1,2,3,4,5]=>[1,6,2,7,3,8,4,10,9,5]=>[3,2,2,2,1]
[5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>[9,10,6,7,8,1,2,3,4,5]=>[1,2,6,3,9,7,4,10,8,5]=>[3,3,2,1,1]
[5,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10]]=>[10,9,6,7,8,1,2,3,4,5]=>[1,2,6,3,7,4,10,9,8,5]=>[4,2,2,1,1]
[5,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10]]=>[10,8,9,6,7,1,2,3,4,5]=>[1,2,3,8,6,4,10,9,7,5]=>[4,3,1,1,1]
[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]=>[1,2,3,6,4,10,9,8,7,5]=>[5,2,1,1,1]
[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]=>[1,2,3,4,10,9,8,7,6,5]=>[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]=>[5,1,6,2,9,7,3,10,8,4]=>[3,3,2,2]
[4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>[10,9,5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,10,9,8,4]=>[4,2,2,2]
[4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>[8,9,10,5,6,7,1,2,3,4]=>[1,8,5,2,9,6,3,10,7,4]=>[3,3,3,1]
[4,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10]]=>[10,8,9,5,6,7,1,2,3,4]=>[1,5,2,8,6,3,10,9,7,4]=>[4,3,2,1]
[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]=>[1,5,2,6,3,10,9,8,7,4]=>[5,2,2,1]
[4,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,1,2,3,4]=>[1,2,9,7,5,3,10,8,6,4]=>[4,4,1,1]
[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]=>[1,2,7,5,3,10,9,8,6,4]=>[5,3,1,1]
[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]=>[1,2,5,3,10,9,8,7,6,4]=>[6,2,1,1]
[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]=>[1,2,3,10,9,8,7,6,5,4]=>[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]=>[7,4,1,8,5,2,10,9,6,3]=>[4,3,3]
[3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>[9,10,7,8,4,5,6,1,2,3]=>[4,1,9,7,5,2,10,8,6,3]=>[4,4,2]
[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]=>[4,1,7,5,2,10,9,8,6,3]=>[5,3,2]
[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]=>[4,1,5,2,10,9,8,7,6,3]=>[6,2,2]
[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]=>[1,8,6,4,2,10,9,7,5,3]=>[5,4,1]
[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]=>[1,6,4,2,10,9,8,7,5,3]=>[6,3,1]
[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]=>[1,4,2,10,9,8,7,6,5,3]=>[7,2,1]
[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]=>[1,2,10,9,8,7,6,5,4,3]=>[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]=>[9,7,5,3,1,10,8,6,4,2]=>[5,5]
[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]=>[7,5,3,1,10,9,8,6,4,2]=>[6,4]
[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]=>[5,3,1,10,9,8,7,6,4,2]=>[7,3]
[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]=>[3,1,10,9,8,7,6,5,4,2]=>[8,2]
[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]=>[1,10,9,8,7,6,5,4,3,2]=>[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]=>[10,9,8,7,6,5,4,3,2,1]=>[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]=>[1,2,3,4,5,6,7,8,9,10,11,12]=>[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]=>[7,1,8,2,9,3,10,4,11,5,12,6]=>[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]=>[12,11,10,9,8,7,6,5,4,3,2,1]=>[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
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
See Mp00067Foata bijection.
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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