Identifier
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Images
=>
Cc0002;cc-rep-0Cc0005;cc-rep-1Cc0002;cc-rep-3
[1]=>[1,0]=>[1]=>[1]
[2]=>[1,0,1,0]=>[2,1]=>[2]
[1,1]=>[1,1,0,0]=>[1,2]=>[1,1]
[3]=>[1,0,1,0,1,0]=>[3,2,1]=>[3]
[2,1]=>[1,0,1,1,0,0]=>[2,3,1]=>[2,1]
[1,1,1]=>[1,1,0,1,0,0]=>[2,1,3]=>[2,1]
[4]=>[1,0,1,0,1,0,1,0]=>[4,3,2,1]=>[4]
[3,1]=>[1,0,1,0,1,1,0,0]=>[3,4,2,1]=>[3,1]
[2,2]=>[1,1,1,0,0,0]=>[1,2,3]=>[1,1,1]
[2,1,1]=>[1,0,1,1,0,1,0,0]=>[3,2,4,1]=>[3,1]
[1,1,1,1]=>[1,1,0,1,0,1,0,0]=>[3,2,1,4]=>[3,1]
[5]=>[1,0,1,0,1,0,1,0,1,0]=>[5,4,3,2,1]=>[5]
[4,1]=>[1,0,1,0,1,0,1,1,0,0]=>[4,5,3,2,1]=>[4,1]
[3,2]=>[1,0,1,1,1,0,0,0]=>[2,3,4,1]=>[2,1,1]
[3,1,1]=>[1,0,1,0,1,1,0,1,0,0]=>[4,3,5,2,1]=>[4,1]
[2,2,1]=>[1,1,1,0,0,1,0,0]=>[3,1,2,4]=>[2,1,1]
[2,1,1,1]=>[1,0,1,1,0,1,0,1,0,0]=>[4,3,2,5,1]=>[4,1]
[1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,0]=>[4,3,2,1,5]=>[4,1]
[6]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[6,5,4,3,2,1]=>[6]
[5,1]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[5,6,4,3,2,1]=>[5,1]
[4,2]=>[1,0,1,0,1,1,1,0,0,0]=>[3,4,5,2,1]=>[3,1,1]
[4,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,0]=>[5,4,6,3,2,1]=>[5,1]
[3,3]=>[1,1,1,0,1,0,0,0]=>[2,1,3,4]=>[2,1,1]
[3,2,1]=>[1,0,1,1,1,0,0,1,0,0]=>[4,2,3,5,1]=>[3,1,1]
[3,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,0]=>[5,4,3,6,2,1]=>[5,1]
[2,2,2]=>[1,1,1,1,0,0,0,0]=>[1,2,3,4]=>[1,1,1,1]
[2,2,1,1]=>[1,1,1,0,0,1,0,1,0,0]=>[4,3,1,2,5]=>[3,1,1]
[2,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,0]=>[5,4,3,2,6,1]=>[5,1]
[1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,0]=>[5,4,3,2,1,6]=>[5,1]
[7]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[7,6,5,4,3,2,1]=>[7]
[6,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[6,7,5,4,3,2,1]=>[6,1]
[5,2]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[4,5,6,3,2,1]=>[4,1,1]
[5,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[6,5,7,4,3,2,1]=>[6,1]
[4,3]=>[1,0,1,1,1,0,1,0,0,0]=>[3,2,4,5,1]=>[3,1,1]
[4,2,1]=>[1,0,1,0,1,1,1,0,0,1,0,0]=>[5,3,4,6,2,1]=>[4,1,1]
[4,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[6,5,4,7,3,2,1]=>[6,1]
[3,3,1]=>[1,1,1,0,1,0,0,1,0,0]=>[4,2,1,3,5]=>[3,1,1]
[3,2,2]=>[1,0,1,1,1,1,0,0,0,0]=>[2,3,4,5,1]=>[2,1,1,1]
[3,2,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,0]=>[5,4,2,3,6,1]=>[4,1,1]
[3,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[6,5,4,3,7,2,1]=>[6,1]
[2,2,2,1]=>[1,1,1,1,0,0,0,1,0,0]=>[4,1,2,3,5]=>[2,1,1,1]
[2,2,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,0]=>[5,4,3,1,2,6]=>[4,1,1]
[2,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[6,5,4,3,2,7,1]=>[6,1]
[1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[6,5,4,3,2,1,7]=>[6,1]
[8]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[8,7,6,5,4,3,2,1]=>[8]
[7,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[7,8,6,5,4,3,2,1]=>[7,1]
[6,2]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[5,6,7,4,3,2,1]=>[5,1,1]
[6,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[7,6,8,5,4,3,2,1]=>[7,1]
[5,3]=>[1,0,1,0,1,1,1,0,1,0,0,0]=>[4,3,5,6,2,1]=>[4,1,1]
[5,2,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[6,4,5,7,3,2,1]=>[5,1,1]
[5,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[7,6,5,8,4,3,2,1]=>[7,1]
[4,4]=>[1,1,1,0,1,0,1,0,0,0]=>[3,2,1,4,5]=>[3,1,1]
[4,3,1]=>[1,0,1,1,1,0,1,0,0,1,0,0]=>[5,3,2,4,6,1]=>[4,1,1]
[4,2,2]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[3,4,5,6,2,1]=>[3,1,1,1]
[4,2,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[6,5,3,4,7,2,1]=>[5,1,1]
[4,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[7,6,5,4,8,3,2,1]=>[7,1]
[3,3,2]=>[1,1,1,0,1,1,0,0,0,0]=>[2,3,1,4,5]=>[2,1,1,1]
[3,3,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,0]=>[5,4,2,1,3,6]=>[4,1,1]
[3,2,2,1]=>[1,0,1,1,1,1,0,0,0,1,0,0]=>[5,2,3,4,6,1]=>[3,1,1,1]
[3,2,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[6,5,4,2,3,7,1]=>[5,1,1]
[3,1,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[7,6,5,4,3,8,2,1]=>[7,1]
[2,2,2,2]=>[1,1,1,1,0,1,0,0,0,0]=>[2,1,3,4,5]=>[2,1,1,1]
[2,2,2,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,0]=>[5,4,1,2,3,6]=>[3,1,1,1]
[2,2,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[6,5,4,3,1,2,7]=>[5,1,1]
[2,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[7,6,5,4,3,2,8,1]=>[7,1]
[1,1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[7,6,5,4,3,2,1,8]=>[7,1]
[9]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[9,8,7,6,5,4,3,2,1]=>[9]
[8,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[8,9,7,6,5,4,3,2,1]=>[8,1]
[7,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[6,7,8,5,4,3,2,1]=>[6,1,1]
[7,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[8,7,9,6,5,4,3,2,1]=>[8,1]
[6,3]=>[1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[5,4,6,7,3,2,1]=>[5,1,1]
[6,2,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[7,5,6,8,4,3,2,1]=>[6,1,1]
[6,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[8,7,6,9,5,4,3,2,1]=>[8,1]
[5,4]=>[1,0,1,1,1,0,1,0,1,0,0,0]=>[4,3,2,5,6,1]=>[4,1,1]
[5,3,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[6,4,3,5,7,2,1]=>[5,1,1]
[5,2,2]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[4,5,6,7,3,2,1]=>[4,1,1,1]
[5,2,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[7,6,4,5,8,3,2,1]=>[6,1,1]
[5,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,9,4,3,2,1]=>[8,1]
[4,4,1]=>[1,1,1,0,1,0,1,0,0,1,0,0]=>[5,3,2,1,4,6]=>[4,1,1]
[4,3,2]=>[1,0,1,1,1,0,1,1,0,0,0,0]=>[3,4,2,5,6,1]=>[3,1,1,1]
[4,3,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[6,5,3,2,4,7,1]=>[5,1,1]
[4,2,2,1]=>[1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[6,3,4,5,7,2,1]=>[4,1,1,1]
[4,2,1,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[7,6,5,3,4,8,2,1]=>[6,1,1]
[4,1,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,4,9,3,2,1]=>[8,1]
[3,3,3]=>[1,1,1,1,1,0,0,0,0,0]=>[1,2,3,4,5]=>[1,1,1,1,1]
[3,3,2,1]=>[1,1,1,0,1,1,0,0,0,1,0,0]=>[5,2,3,1,4,6]=>[3,1,1,1]
[3,3,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[6,5,4,2,1,3,7]=>[5,1,1]
[3,2,2,2]=>[1,0,1,1,1,1,0,1,0,0,0,0]=>[3,2,4,5,6,1]=>[3,1,1,1]
[3,2,2,1,1]=>[1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[6,5,2,3,4,7,1]=>[4,1,1,1]
[3,2,1,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[7,6,5,4,2,3,8,1]=>[6,1,1]
[3,1,1,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,4,3,9,2,1]=>[8,1]
[2,2,2,2,1]=>[1,1,1,1,0,1,0,0,0,1,0,0]=>[5,2,1,3,4,6]=>[3,1,1,1]
[2,2,2,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[6,5,4,1,2,3,7]=>[4,1,1,1]
[2,2,1,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[7,6,5,4,3,1,2,8]=>[6,1,1]
[2,1,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,4,3,2,9,1]=>[8,1]
[1,1,1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,4,3,2,1,9]=>[8,1]
[10]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[10,9,8,7,6,5,4,3,2,1]=>[10]
[9,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[9,10,8,7,6,5,4,3,2,1]=>[9,1]
[8,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[7,8,9,6,5,4,3,2,1]=>[7,1,1]
[8,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[9,8,10,7,6,5,4,3,2,1]=>[9,1]
[7,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[6,5,7,8,4,3,2,1]=>[6,1,1]
[7,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[8,6,7,9,5,4,3,2,1]=>[7,1,1]
[7,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[9,8,7,10,6,5,4,3,2,1]=>[9,1]
[6,4]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[5,4,3,6,7,2,1]=>[5,1,1]
[6,3,1]=>[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[7,5,4,6,8,3,2,1]=>[6,1,1]
[6,2,2]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[5,6,7,8,4,3,2,1]=>[5,1,1,1]
[6,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[8,7,5,6,9,4,3,2,1]=>[7,1,1]
[6,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,10,5,4,3,2,1]=>[9,1]
[5,5]=>[1,1,1,0,1,0,1,0,1,0,0,0]=>[4,3,2,1,5,6]=>[4,1,1]
[5,4,1]=>[1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[6,4,3,2,5,7,1]=>[5,1,1]
[5,3,2]=>[1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[4,5,3,6,7,2,1]=>[4,1,1,1]
[5,3,1,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[7,6,4,3,5,8,2,1]=>[6,1,1]
[5,2,2,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[7,4,5,6,8,3,2,1]=>[5,1,1,1]
[5,2,1,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[8,7,6,4,5,9,3,2,1]=>[7,1,1]
[5,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,5,10,4,3,2,1]=>[9,1]
[4,4,2]=>[1,1,1,0,1,0,1,1,0,0,0,0]=>[3,4,2,1,5,6]=>[3,1,1,1]
[4,4,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[6,5,3,2,1,4,7]=>[5,1,1]
[4,3,3]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[2,3,4,5,6,1]=>[2,1,1,1,1]
[4,3,2,1]=>[1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[6,3,4,2,5,7,1]=>[4,1,1,1]
[4,3,1,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[7,6,5,3,2,4,8,1]=>[6,1,1]
[4,2,2,2]=>[1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[4,3,5,6,7,2,1]=>[4,1,1,1]
[4,2,2,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[7,6,3,4,5,8,2,1]=>[5,1,1,1]
[4,2,1,1,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,3,4,9,2,1]=>[7,1,1]
[4,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,5,4,10,3,2,1]=>[9,1]
[3,3,3,1]=>[1,1,1,1,1,0,0,0,0,1,0,0]=>[5,1,2,3,4,6]=>[2,1,1,1,1]
[3,3,2,2]=>[1,1,1,0,1,1,0,1,0,0,0,0]=>[3,2,4,1,5,6]=>[3,1,1,1]
[3,3,2,1,1]=>[1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[6,5,2,3,1,4,7]=>[4,1,1,1]
[3,3,1,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[7,6,5,4,2,1,3,8]=>[6,1,1]
[3,2,2,2,1]=>[1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[6,3,2,4,5,7,1]=>[4,1,1,1]
[3,2,2,1,1,1]=>[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[7,6,5,2,3,4,8,1]=>[5,1,1,1]
[3,2,1,1,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,4,2,3,9,1]=>[7,1,1]
[3,1,1,1,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,5,4,3,10,2,1]=>[9,1]
[2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[3,2,1,4,5,6]=>[3,1,1,1]
[2,2,2,2,1,1]=>[1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[6,5,2,1,3,4,7]=>[4,1,1,1]
[2,2,2,1,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]=>[7,6,5,4,1,2,3,8]=>[5,1,1,1]
[2,2,1,1,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,4,3,1,2,9]=>[7,1,1]
[2,1,1,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,5,4,3,2,10,1]=>[9,1]
[1,1,1,1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,5,4,3,2,1,10]=>[9,1]
[11]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[11,10,9,8,7,6,5,4,3,2,1]=>[11]
[10,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[10,11,9,8,7,6,5,4,3,2,1]=>[10,1]
[9,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[8,9,10,7,6,5,4,3,2,1]=>[8,1,1]
[7,4]=>[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[6,5,4,7,8,3,2,1]=>[6,1,1]
[7,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[6,7,8,9,5,4,3,2,1]=>[6,1,1,1]
[6,5]=>[1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[5,4,3,2,6,7,1]=>[5,1,1]
[6,4,1]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[7,5,4,3,6,8,2,1]=>[6,1,1]
[6,3,2]=>[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[5,6,4,7,8,3,2,1]=>[5,1,1,1]
[5,5,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[6,4,3,2,1,5,7]=>[5,1,1]
[5,4,2]=>[1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[4,5,3,2,6,7,1]=>[4,1,1,1]
[5,3,3]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[3,4,5,6,7,2,1]=>[3,1,1,1,1]
[5,2,2,2]=>[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[5,4,6,7,8,3,2,1]=>[5,1,1,1]
[4,4,3]=>[1,1,1,0,1,1,1,0,0,0,0,0]=>[2,3,4,1,5,6]=>[2,1,1,1,1]
[4,4,2,1]=>[1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>[6,3,4,2,1,5,7]=>[4,1,1,1]
[4,4,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[7,6,5,3,2,1,4,8]=>[6,1,1]
[4,3,3,1]=>[1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>[6,2,3,4,5,7,1]=>[3,1,1,1,1]
[4,3,2,2]=>[1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>[4,3,5,2,6,7,1]=>[4,1,1,1]
[3,3,3,2]=>[1,1,1,1,1,0,0,1,0,0,0,0]=>[3,1,2,4,5,6]=>[2,1,1,1,1]
[3,3,3,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[6,5,1,2,3,4,7]=>[3,1,1,1,1]
[3,3,2,2,1]=>[1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[6,3,2,4,1,5,7]=>[4,1,1,1]
[3,3,2,1,1,1]=>[1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]=>[7,6,5,2,3,1,4,8]=>[5,1,1,1]
[3,3,1,1,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,4,2,1,3,9]=>[7,1,1]
[3,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[4,3,2,5,6,7,1]=>[4,1,1,1]
[3,2,2,2,1,1]=>[1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[7,6,3,2,4,5,8,1]=>[5,1,1,1]
[2,2,2,2,2,1]=>[1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[6,3,2,1,4,5,7]=>[4,1,1,1]
[2,2,2,2,1,1,1]=>[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]=>[7,6,5,2,1,3,4,8]=>[5,1,1,1]
[2,2,2,1,1,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,4,1,2,3,9]=>[6,1,1,1]
[2,2,1,1,1,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,5,4,3,1,2,10]=>[8,1,1]
[2,1,1,1,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[10,9,8,7,6,5,4,3,2,11,1]=>[10,1]
[1,1,1,1,1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[10,9,8,7,6,5,4,3,2,1,11]=>[10,1]
[8,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[7,8,9,10,6,5,4,3,2,1]=>[7,1,1,1]
[7,5]=>[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[6,5,4,3,7,8,2,1]=>[6,1,1]
[6,6]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[5,4,3,2,1,6,7]=>[5,1,1]
[6,5,1]=>[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[7,5,4,3,2,6,8,1]=>[6,1,1]
[6,3,3]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[4,5,6,7,8,3,2,1]=>[4,1,1,1,1]
[5,5,2]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[4,5,3,2,1,6,7]=>[4,1,1,1]
[5,5,1,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>[7,6,4,3,2,1,5,8]=>[6,1,1]
[5,4,3]=>[1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[3,4,5,2,6,7,1]=>[3,1,1,1,1]
[5,3,3,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>[7,3,4,5,6,8,2,1]=>[4,1,1,1,1]
[4,4,4]=>[1,1,1,1,1,0,1,0,0,0,0,0]=>[2,1,3,4,5,6]=>[2,1,1,1,1]
[4,4,3,1]=>[1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>[6,2,3,4,1,5,7]=>[3,1,1,1,1]
[4,4,2,2]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[4,3,5,2,1,6,7]=>[4,1,1,1]
[4,4,2,1,1]=>[1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]=>[7,6,3,4,2,1,5,8]=>[5,1,1,1]
[4,4,1,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,3,2,1,4,9]=>[7,1,1]
[4,3,3,2]=>[1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[4,2,3,5,6,7,1]=>[3,1,1,1,1]
[4,3,3,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[7,6,2,3,4,5,8,1]=>[4,1,1,1,1]
[4,3,2,2,1]=>[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[7,4,3,5,2,6,8,1]=>[5,1,1,1]
[4,2,2,2,2]=>[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[5,4,3,6,7,8,2,1]=>[5,1,1,1]
[3,3,3,3]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,3,4,5,6]=>[1,1,1,1,1,1]
[3,3,3,2,1]=>[1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[6,3,1,2,4,5,7]=>[3,1,1,1,1]
[3,3,3,1,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]=>[7,6,5,1,2,3,4,8]=>[4,1,1,1,1]
[3,3,2,2,2]=>[1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[4,3,2,5,1,6,7]=>[4,1,1,1]
[3,3,2,2,1,1]=>[1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]=>[7,6,3,2,4,1,5,8]=>[5,1,1,1]
[3,3,2,1,1,1,1]=>[1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,0]=>[8,7,6,5,2,3,1,4,9]=>[6,1,1,1]
[3,3,1,1,1,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,5,4,2,1,3,10]=>[8,1,1]
[2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[4,3,2,1,5,6,7]=>[4,1,1,1]
[2,2,2,2,2,1,1]=>[1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]=>[7,6,3,2,1,4,5,8]=>[5,1,1,1]
[7,6]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[6,5,4,3,2,7,8,1]=>[6,1,1]
[7,3,3]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[5,6,7,8,9,4,3,2,1]=>[5,1,1,1,1]
[6,6,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[7,5,4,3,2,1,6,8]=>[6,1,1]
[5,5,3]=>[1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>[3,4,5,2,1,6,7]=>[3,1,1,1,1]
[5,5,2,1]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]=>[7,4,5,3,2,1,6,8]=>[5,1,1,1]
[5,5,1,1,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[8,7,6,4,3,2,1,5,9]=>[7,1,1]
[5,4,4]=>[1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[3,2,4,5,6,7,1]=>[3,1,1,1,1]
[5,4,2,2]=>[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[5,4,6,3,2,7,8,1]=>[5,1,1,1]
[4,4,4,1]=>[1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[6,2,1,3,4,5,7]=>[3,1,1,1,1]
[4,4,3,2]=>[1,1,1,0,1,1,1,0,0,1,0,0,0,0]=>[4,2,3,5,1,6,7]=>[3,1,1,1,1]
[4,4,3,1,1]=>[1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]=>[7,6,2,3,4,1,5,8]=>[4,1,1,1,1]
[4,4,2,2,1]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]=>[7,4,3,5,2,1,6,8]=>[5,1,1,1]
[4,4,2,1,1,1]=>[1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0]=>[8,7,6,3,4,2,1,5,9]=>[6,1,1,1]
[4,4,1,1,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,5,3,2,1,4,10]=>[8,1,1]
[4,3,3,3]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,3,4,5,6,7,1]=>[2,1,1,1,1,1]
[4,3,3,2,1]=>[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[7,4,2,3,5,6,8,1]=>[4,1,1,1,1]
[4,3,2,2,2]=>[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[5,4,3,6,2,7,8,1]=>[5,1,1,1]
[3,3,3,3,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[6,1,2,3,4,5,7]=>[2,1,1,1,1,1]
[3,3,3,2,2]=>[1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[4,3,1,2,5,6,7]=>[3,1,1,1,1]
[3,3,1,1,1,1,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[10,9,8,7,6,5,4,2,1,3,11]=>[9,1,1]
[3,2,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[5,4,3,2,6,7,8,1]=>[5,1,1,1]
[2,2,2,2,2,2,1]=>[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]=>[7,4,3,2,1,5,6,8]=>[5,1,1,1]
[8,3,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[6,7,8,9,10,5,4,3,2,1]=>[6,1,1,1,1]
[7,7]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[6,5,4,3,2,1,7,8]=>[6,1,1]
[7,6,1]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[8,6,5,4,3,2,7,9,1]=>[7,1,1]
[6,6,2]=>[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[5,6,4,3,2,1,7,8]=>[5,1,1,1]
[6,6,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>[8,7,5,4,3,2,1,6,9]=>[7,1,1]
[6,5,3]=>[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>[4,5,6,3,2,7,8,1]=>[4,1,1,1,1]
[6,4,4]=>[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[4,3,5,6,7,8,2,1]=>[4,1,1,1,1]
[5,5,4]=>[1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>[3,2,4,5,1,6,7]=>[3,1,1,1,1]
[5,5,3,1]=>[1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]=>[7,3,4,5,2,1,6,8]=>[4,1,1,1,1]
[5,5,2,2]=>[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]=>[5,4,6,3,2,1,7,8]=>[5,1,1,1]
[5,5,2,1,1]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0]=>[8,7,4,5,3,2,1,6,9]=>[6,1,1,1]
[5,5,1,1,1,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,4,3,2,1,5,10]=>[8,1,1]
[5,4,4,1]=>[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[7,3,2,4,5,6,8,1]=>[4,1,1,1,1]
[5,3,3,3]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[3,4,5,6,7,8,2,1]=>[3,1,1,1,1,1]
[4,4,4,2]=>[1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>[4,2,1,3,5,6,7]=>[3,1,1,1,1]
[4,4,3,3]=>[1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[2,3,4,5,1,6,7]=>[2,1,1,1,1,1]
[4,4,3,2,1]=>[1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]=>[7,4,2,3,5,1,6,8]=>[4,1,1,1,1]
[4,4,2,2,2]=>[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]=>[5,4,3,6,2,1,7,8]=>[5,1,1,1]
[4,4,1,1,1,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[10,9,8,7,6,5,3,2,1,4,11]=>[9,1,1]
[4,3,3,3,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[7,2,3,4,5,6,8,1]=>[3,1,1,1,1,1]
[4,3,3,2,2]=>[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[5,4,2,3,6,7,8,1]=>[4,1,1,1,1]
[3,3,3,3,2]=>[1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[4,1,2,3,5,6,7]=>[2,1,1,1,1,1]
[3,3,3,3,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]=>[7,6,1,2,3,4,5,8]=>[3,1,1,1,1,1]
[3,3,3,2,2,1]=>[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]=>[7,4,3,1,2,5,6,8]=>[4,1,1,1,1]
[3,3,2,2,2,2]=>[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]=>[5,4,3,2,6,1,7,8]=>[5,1,1,1]
[2,2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[5,4,3,2,1,6,7,8]=>[5,1,1,1]
[8,7]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[7,6,5,4,3,2,8,9,1]=>[7,1,1]
[7,7,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[8,6,5,4,3,2,1,7,9]=>[7,1,1]
[6,6,3]=>[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]=>[4,5,6,3,2,1,7,8]=>[4,1,1,1,1]
[6,6,2,1]=>[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0,0]=>[8,5,6,4,3,2,1,7,9]=>[6,1,1,1]
[6,6,1,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[9,8,7,5,4,3,2,1,6,10]=>[8,1,1]
[6,5,4]=>[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>[4,3,5,6,2,7,8,1]=>[4,1,1,1,1]
[6,3,3,3]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[4,5,6,7,8,9,3,2,1]=>[4,1,1,1,1,1]
[5,5,5]=>[1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[3,2,1,4,5,6,7]=>[3,1,1,1,1]
[5,4,4,2]=>[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>[5,3,2,4,6,7,8,1]=>[4,1,1,1,1]
[5,4,3,3]=>[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[3,4,5,6,2,7,8,1]=>[3,1,1,1,1,1]
[4,4,4,3]=>[1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[2,3,1,4,5,6,7]=>[2,1,1,1,1,1]
[4,4,3,3,1]=>[1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]=>[7,2,3,4,5,1,6,8]=>[3,1,1,1,1,1]
[4,3,3,3,2]=>[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[5,2,3,4,6,7,8,1]=>[3,1,1,1,1,1]
[3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[2,1,3,4,5,6,7]=>[2,1,1,1,1,1]
[3,3,3,3,2,1]=>[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]=>[7,4,1,2,3,5,6,8]=>[3,1,1,1,1,1]
[3,3,3,2,2,2]=>[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]=>[5,4,3,1,2,6,7,8]=>[4,1,1,1,1]
[3,2,2,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[6,5,4,3,2,7,8,9,1]=>[6,1,1,1]
[8,8]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[7,6,5,4,3,2,1,8,9]=>[7,1,1]
[7,7,2]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[6,7,5,4,3,2,1,8,9]=>[6,1,1,1]
[7,7,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>[9,8,6,5,4,3,2,1,7,10]=>[8,1,1]
[7,3,3,3]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[5,6,7,8,9,10,4,3,2,1]=>[5,1,1,1,1,1]
[6,6,1,1,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[10,9,8,7,5,4,3,2,1,6,11]=>[9,1,1]
[6,5,5]=>[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[4,3,2,5,6,7,8,1]=>[4,1,1,1,1]
[5,5,5,1]=>[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]=>[7,3,2,1,4,5,6,8]=>[4,1,1,1,1]
[5,5,4,2]=>[1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]=>[5,3,2,4,6,1,7,8]=>[4,1,1,1,1]
[5,4,4,3]=>[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[3,4,2,5,6,7,8,1]=>[3,1,1,1,1,1]
[4,4,4,4]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,2,3,4,5,6,7]=>[1,1,1,1,1,1,1]
[4,4,4,3,1]=>[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]=>[7,2,3,1,4,5,6,8]=>[3,1,1,1,1,1]
[4,4,4,2,2]=>[1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]=>[5,4,2,1,3,6,7,8]=>[4,1,1,1,1]
[4,4,3,3,2]=>[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]=>[5,2,3,4,6,1,7,8]=>[3,1,1,1,1,1]
[4,3,3,3,3]=>[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[3,2,4,5,6,7,8,1]=>[3,1,1,1,1,1]
[4,3,3,2,2,2]=>[1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]=>[6,5,4,2,3,7,8,9,1]=>[5,1,1,1,1]
[3,3,3,3,3,1]=>[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]=>[7,2,1,3,4,5,6,8]=>[3,1,1,1,1,1]
[3,3,3,3,2,2]=>[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]=>[5,4,1,2,3,6,7,8]=>[3,1,1,1,1,1]
[3,3,3,3,1,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]=>[9,8,7,6,1,2,3,4,5,10]=>[5,1,1,1,1,1]
[3,3,2,2,2,2,2]=>[1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[6,5,4,3,2,7,1,8,9]=>[6,1,1,1]
[2,2,2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[6,5,4,3,2,1,7,8,9]=>[6,1,1,1]
[9,8]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[8,7,6,5,4,3,2,9,10,1]=>[8,1,1]
[8,8,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[9,7,6,5,4,3,2,1,8,10]=>[8,1,1]
[7,7,1,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[10,9,8,6,5,4,3,2,1,7,11]=>[9,1,1]
[6,6,5]=>[1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]=>[4,3,2,5,6,1,7,8]=>[4,1,1,1,1]
[5,5,5,2]=>[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]=>[5,3,2,1,4,6,7,8]=>[4,1,1,1,1]
[5,5,4,3]=>[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]=>[3,4,2,5,6,1,7,8]=>[3,1,1,1,1,1]
[5,4,4,4]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,1]=>[2,1,1,1,1,1,1]
[4,4,4,4,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]=>[7,1,2,3,4,5,6,8]=>[2,1,1,1,1,1,1]
[4,4,4,3,2]=>[1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]=>[5,2,3,1,4,6,7,8]=>[3,1,1,1,1,1]
[4,4,3,3,3]=>[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]=>[3,2,4,5,6,1,7,8]=>[3,1,1,1,1,1]
[3,3,3,3,3,2]=>[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]=>[5,2,1,3,4,6,7,8]=>[3,1,1,1,1,1]
[3,3,3,2,2,2,2]=>[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]=>[6,5,4,3,1,2,7,8,9]=>[5,1,1,1,1]
[3,2,2,2,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[7,6,5,4,3,2,8,9,10,1]=>[7,1,1,1]
[]=>[]=>[]=>[]
[4,4,4,4,3,1]=>[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0]=>[8,3,1,2,4,5,6,7,9]=>[3,1,1,1,1,1,1]
[5,4,4,4,3]=>[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]=>[4,2,3,5,6,7,8,9,1]=>[3,1,1,1,1,1,1]
[4,4,4,4,2]=>[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]=>[5,1,2,3,4,6,7,8]=>[2,1,1,1,1,1,1]
[2,2,2,2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[7,6,5,4,3,2,1,8,9,10]=>[7,1,1,1]
[3,3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]=>[3,2,1,4,5,6,7,8]=>[3,1,1,1,1,1]
[3,3,3,3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[5,4,3,2,1,6,7,8,9,10]=>[5,1,1,1,1,1]
[3,3,3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]=>[4,3,2,1,5,6,7,8,9]=>[4,1,1,1,1,1]
[4,3,3,3,3,3]=>[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]=>[4,3,2,5,6,7,8,9,1]=>[4,1,1,1,1,1]
[4,4,4,4,4]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,2,3,4,5,6,7,8]=>[1,1,1,1,1,1,1,1]
[4,4,4,3,3,3]=>[1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0]=>[4,3,2,5,1,6,7,8,9]=>[4,1,1,1,1,1]
[4,4,4,4,4,4]=>[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]=>[2,1,3,4,5,6,7,8,9]=>[2,1,1,1,1,1,1,1]
[5,5,5,5]=>[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]=>[2,1,3,4,5,6,7,8]=>[2,1,1,1,1,1,1]
[6,6,6,6]=>[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]=>[3,2,1,4,5,6,7,8,9]=>[3,1,1,1,1,1,1]
[6,6,6]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[4,3,2,1,5,6,7,8]=>[4,1,1,1,1]
[7,6,6]=>[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[5,4,3,2,6,7,8,9,1]=>[5,1,1,1,1]
[10,10]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[9,8,7,6,5,4,3,2,1,10,11]=>[9,1,1]
[6,6,6,6,6]=>[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]=>[2,1,3,4,5,6,7,8,9,10]=>[2,1,1,1,1,1,1,1,1]
[5,5,5,5,5,5]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[1,2,3,4,5,6,7,8,9,10]=>[1,1,1,1,1,1,1,1,1,1]
[7,7,7,7]=>[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]=>[4,3,2,1,5,6,7,8,9,10]=>[4,1,1,1,1,1,1]
[7,7,7]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]=>[5,4,3,2,1,6,7,8,9]=>[5,1,1,1,1]
[5,5,5,5,5]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[1,2,3,4,5,6,7,8,9]=>[1,1,1,1,1,1,1,1,1]
[9,9]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[8,7,6,5,4,3,2,1,9,10]=>[8,1,1]
[8,8,8]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]=>[6,5,4,3,2,1,7,8,9,10]=>[6,1,1,1,1]
[4,4,4,4,4,4,4]=>[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]=>[3,2,1,4,5,6,7,8,9,10]=>[3,1,1,1,1,1,1,1]
[6,5,5,5]=>[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]=>[3,2,4,5,6,7,8,9,1]=>[3,1,1,1,1,1,1]
[5,4,4,4,4]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[2,3,4,5,6,7,8,9,1]=>[2,1,1,1,1,1,1,1]
[4,4,4,4,4,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]=>[8,1,2,3,4,5,6,7,9]=>[2,1,1,1,1,1,1,1]
[5,5,5,5,1]=>[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]=>[8,2,1,3,4,5,6,7,9]=>[3,1,1,1,1,1,1]
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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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