Identifier
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Images
=>
Cc0002;cc-rep-0Cc0005;cc-rep-1
[1]=>[1,0]=>[1]=>[1]
[2]=>[1,0,1,0]=>[1,2]=>[1,2]
[1,1]=>[1,1,0,0]=>[2,1]=>[2,1]
[3]=>[1,0,1,0,1,0]=>[1,2,3]=>[1,2,3]
[2,1]=>[1,0,1,1,0,0]=>[1,3,2]=>[3,1,2]
[1,1,1]=>[1,1,0,1,0,0]=>[2,3,1]=>[2,3,1]
[4]=>[1,0,1,0,1,0,1,0]=>[1,2,3,4]=>[1,2,3,4]
[3,1]=>[1,0,1,0,1,1,0,0]=>[1,2,4,3]=>[4,1,2,3]
[2,2]=>[1,1,1,0,0,0]=>[3,2,1]=>[3,2,1]
[2,1,1]=>[1,0,1,1,0,1,0,0]=>[1,3,4,2]=>[3,4,1,2]
[1,1,1,1]=>[1,1,0,1,0,1,0,0]=>[2,3,4,1]=>[2,3,4,1]
[5]=>[1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5]=>[1,2,3,4,5]
[4,1]=>[1,0,1,0,1,0,1,1,0,0]=>[1,2,3,5,4]=>[5,1,2,3,4]
[3,2]=>[1,0,1,1,1,0,0,0]=>[1,4,3,2]=>[4,3,1,2]
[3,1,1]=>[1,0,1,0,1,1,0,1,0,0]=>[1,2,4,5,3]=>[4,5,1,2,3]
[2,2,1]=>[1,1,1,0,0,1,0,0]=>[3,2,4,1]=>[3,2,4,1]
[2,1,1,1]=>[1,0,1,1,0,1,0,1,0,0]=>[1,3,4,5,2]=>[3,4,5,1,2]
[1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,0]=>[2,3,4,5,1]=>[2,3,4,5,1]
[6]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]
[5,1]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,6,5]=>[6,1,2,3,4,5]
[4,2]=>[1,0,1,0,1,1,1,0,0,0]=>[1,2,5,4,3]=>[5,4,1,2,3]
[4,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,0]=>[1,2,3,5,6,4]=>[5,6,1,2,3,4]
[3,3]=>[1,1,1,0,1,0,0,0]=>[3,4,2,1]=>[3,4,2,1]
[3,2,1]=>[1,0,1,1,1,0,0,1,0,0]=>[1,4,3,5,2]=>[4,3,5,1,2]
[3,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,0]=>[1,2,4,5,6,3]=>[4,5,6,1,2,3]
[2,2,2]=>[1,1,1,1,0,0,0,0]=>[4,3,2,1]=>[4,3,2,1]
[2,2,1,1]=>[1,1,1,0,0,1,0,1,0,0]=>[3,2,4,5,1]=>[3,2,4,5,1]
[2,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,0]=>[1,3,4,5,6,2]=>[3,4,5,6,1,2]
[1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,0]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]
[7]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7]
[6,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,7,6]=>[7,1,2,3,4,5,6]
[5,2]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,6,5,4]=>[6,5,1,2,3,4]
[5,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,2,3,4,6,7,5]=>[6,7,1,2,3,4,5]
[4,3]=>[1,0,1,1,1,0,1,0,0,0]=>[1,4,5,3,2]=>[4,5,3,1,2]
[4,2,1]=>[1,0,1,0,1,1,1,0,0,1,0,0]=>[1,2,5,4,6,3]=>[5,4,6,1,2,3]
[4,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[1,2,3,5,6,7,4]=>[5,6,7,1,2,3,4]
[3,3,1]=>[1,1,1,0,1,0,0,1,0,0]=>[3,4,2,5,1]=>[3,4,2,5,1]
[3,2,2]=>[1,0,1,1,1,1,0,0,0,0]=>[1,5,4,3,2]=>[5,4,3,1,2]
[3,2,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,0]=>[1,4,3,5,6,2]=>[4,3,5,6,1,2]
[3,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[1,2,4,5,6,7,3]=>[4,5,6,7,1,2,3]
[2,2,2,1]=>[1,1,1,1,0,0,0,1,0,0]=>[4,3,2,5,1]=>[4,3,2,5,1]
[2,2,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,0]=>[3,2,4,5,6,1]=>[3,2,4,5,6,1]
[2,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[1,3,4,5,6,7,2]=>[3,4,5,6,7,1,2]
[1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[2,3,4,5,6,7,1]=>[2,3,4,5,6,7,1]
[8]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8]
[7,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,6,8,7]=>[8,1,2,3,4,5,6,7]
[6,2]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,7,6,5]=>[7,6,1,2,3,4,5]
[6,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,2,3,4,5,7,8,6]=>[7,8,1,2,3,4,5,6]
[5,3]=>[1,0,1,0,1,1,1,0,1,0,0,0]=>[1,2,5,6,4,3]=>[5,6,4,1,2,3]
[5,2,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[1,2,3,6,5,7,4]=>[6,5,7,1,2,3,4]
[5,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[1,2,3,4,6,7,8,5]=>[6,7,8,1,2,3,4,5]
[4,4]=>[1,1,1,0,1,0,1,0,0,0]=>[3,4,5,2,1]=>[3,4,5,2,1]
[4,3,1]=>[1,0,1,1,1,0,1,0,0,1,0,0]=>[1,4,5,3,6,2]=>[4,5,3,6,1,2]
[4,2,2]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,6,5,4,3]=>[6,5,4,1,2,3]
[4,2,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[1,2,5,4,6,7,3]=>[5,4,6,7,1,2,3]
[4,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[1,2,3,5,6,7,8,4]=>[5,6,7,8,1,2,3,4]
[3,3,2]=>[1,1,1,0,1,1,0,0,0,0]=>[3,5,4,2,1]=>[5,3,4,2,1]
[3,3,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,0]=>[3,4,2,5,6,1]=>[3,4,2,5,6,1]
[3,2,2,1]=>[1,0,1,1,1,1,0,0,0,1,0,0]=>[1,5,4,3,6,2]=>[5,4,3,6,1,2]
[3,2,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[1,4,3,5,6,7,2]=>[4,3,5,6,7,1,2]
[3,1,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[1,2,4,5,6,7,8,3]=>[4,5,6,7,8,1,2,3]
[2,2,2,2]=>[1,1,1,1,0,1,0,0,0,0]=>[4,5,3,2,1]=>[4,5,3,2,1]
[2,2,2,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,0]=>[4,3,2,5,6,1]=>[4,3,2,5,6,1]
[2,2,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[3,2,4,5,6,7,1]=>[3,2,4,5,6,7,1]
[2,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[1,3,4,5,6,7,8,2]=>[3,4,5,6,7,8,1,2]
[1,1,1,1,1,1,1,1]=>[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[2,3,4,5,6,7,8,1]=>[2,3,4,5,6,7,8,1]
[9]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7,8,9]=>[1,2,3,4,5,6,7,8,9]
[8,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,6,7,9,8]=>[9,1,2,3,4,5,6,7,8]
[7,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,5,8,7,6]=>[8,7,1,2,3,4,5,6]
[7,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,2,3,4,5,6,8,9,7]=>[8,9,1,2,3,4,5,6,7]
[6,3]=>[1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[1,2,3,6,7,5,4]=>[6,7,5,1,2,3,4]
[6,2,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[1,2,3,4,7,6,8,5]=>[7,6,8,1,2,3,4,5]
[6,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[1,2,3,4,5,7,8,9,6]=>[7,8,9,1,2,3,4,5,6]
[5,4]=>[1,0,1,1,1,0,1,0,1,0,0,0]=>[1,4,5,6,3,2]=>[4,5,6,3,1,2]
[5,3,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[1,2,5,6,4,7,3]=>[5,6,4,7,1,2,3]
[5,2,2]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,7,6,5,4]=>[7,6,5,1,2,3,4]
[5,2,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[1,2,3,6,5,7,8,4]=>[6,5,7,8,1,2,3,4]
[5,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[1,2,3,4,6,7,8,9,5]=>[6,7,8,9,1,2,3,4,5]
[4,4,1]=>[1,1,1,0,1,0,1,0,0,1,0,0]=>[3,4,5,2,6,1]=>[3,4,5,2,6,1]
[4,3,2]=>[1,0,1,1,1,0,1,1,0,0,0,0]=>[1,4,6,5,3,2]=>[6,4,5,3,1,2]
[4,3,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[1,4,5,3,6,7,2]=>[4,5,3,6,7,1,2]
[4,2,2,1]=>[1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[1,2,6,5,4,7,3]=>[6,5,4,7,1,2,3]
[4,2,1,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[1,2,5,4,6,7,8,3]=>[5,4,6,7,8,1,2,3]
[4,1,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[1,2,3,5,6,7,8,9,4]=>[5,6,7,8,9,1,2,3,4]
[3,3,3]=>[1,1,1,1,1,0,0,0,0,0]=>[5,4,3,2,1]=>[5,4,3,2,1]
[3,3,2,1]=>[1,1,1,0,1,1,0,0,0,1,0,0]=>[3,5,4,2,6,1]=>[5,3,4,2,6,1]
[3,3,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[3,4,2,5,6,7,1]=>[3,4,2,5,6,7,1]
[3,2,2,2]=>[1,0,1,1,1,1,0,1,0,0,0,0]=>[1,5,6,4,3,2]=>[5,6,4,3,1,2]
[3,2,2,1,1]=>[1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[1,5,4,3,6,7,2]=>[5,4,3,6,7,1,2]
[3,2,1,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[1,4,3,5,6,7,8,2]=>[4,3,5,6,7,8,1,2]
[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]=>[1,2,4,5,6,7,8,9,3]=>[4,5,6,7,8,9,1,2,3]
[2,2,2,2,1]=>[1,1,1,1,0,1,0,0,0,1,0,0]=>[4,5,3,2,6,1]=>[4,5,3,2,6,1]
[2,2,2,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[4,3,2,5,6,7,1]=>[4,3,2,5,6,7,1]
[2,2,1,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[3,2,4,5,6,7,8,1]=>[3,2,4,5,6,7,8,1]
[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]=>[1,3,4,5,6,7,8,9,2]=>[3,4,5,6,7,8,9,1,2]
[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]=>[2,3,4,5,6,7,8,9,1]=>[2,3,4,5,6,7,8,9,1]
[10]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7,8,9,10]=>[1,2,3,4,5,6,7,8,9,10]
[9,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,6,7,8,10,9]=>[10,1,2,3,4,5,6,7,8,9]
[8,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,5,6,9,8,7]=>[9,8,1,2,3,4,5,6,7]
[8,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,2,3,4,5,6,7,9,10,8]=>[9,10,1,2,3,4,5,6,7,8]
[7,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[1,2,3,4,7,8,6,5]=>[7,8,6,1,2,3,4,5]
[7,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[1,2,3,4,5,8,7,9,6]=>[8,7,9,1,2,3,4,5,6]
[7,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[1,2,3,4,5,6,8,9,10,7]=>[8,9,10,1,2,3,4,5,6,7]
[6,4]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[1,2,5,6,7,4,3]=>[5,6,7,4,1,2,3]
[6,3,1]=>[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[1,2,3,6,7,5,8,4]=>[6,7,5,8,1,2,3,4]
[6,2,2]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,4,8,7,6,5]=>[8,7,6,1,2,3,4,5]
[6,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[1,2,3,4,7,6,8,9,5]=>[7,6,8,9,1,2,3,4,5]
[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]=>[1,2,3,4,5,7,8,9,10,6]=>[7,8,9,10,1,2,3,4,5,6]
[5,5]=>[1,1,1,0,1,0,1,0,1,0,0,0]=>[3,4,5,6,2,1]=>[3,4,5,6,2,1]
[5,4,1]=>[1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[1,4,5,6,3,7,2]=>[4,5,6,3,7,1,2]
[5,3,2]=>[1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[1,2,5,7,6,4,3]=>[7,5,6,4,1,2,3]
[5,3,1,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[1,2,5,6,4,7,8,3]=>[5,6,4,7,8,1,2,3]
[5,2,2,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[1,2,3,7,6,5,8,4]=>[7,6,5,8,1,2,3,4]
[5,2,1,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[1,2,3,6,5,7,8,9,4]=>[6,5,7,8,9,1,2,3,4]
[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]=>[1,2,3,4,6,7,8,9,10,5]=>[6,7,8,9,10,1,2,3,4,5]
[4,4,2]=>[1,1,1,0,1,0,1,1,0,0,0,0]=>[3,4,6,5,2,1]=>[6,3,4,5,2,1]
[4,4,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[3,4,5,2,6,7,1]=>[3,4,5,2,6,7,1]
[4,3,3]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[1,6,5,4,3,2]=>[6,5,4,3,1,2]
[4,3,2,1]=>[1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[1,4,6,5,3,7,2]=>[6,4,5,3,7,1,2]
[4,3,1,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[1,4,5,3,6,7,8,2]=>[4,5,3,6,7,8,1,2]
[4,2,2,2]=>[1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[1,2,6,7,5,4,3]=>[6,7,5,4,1,2,3]
[4,2,2,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[1,2,6,5,4,7,8,3]=>[6,5,4,7,8,1,2,3]
[4,2,1,1,1,1]=>[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[1,2,5,4,6,7,8,9,3]=>[5,4,6,7,8,9,1,2,3]
[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]=>[1,2,3,5,6,7,8,9,10,4]=>[5,6,7,8,9,10,1,2,3,4]
[3,3,3,1]=>[1,1,1,1,1,0,0,0,0,1,0,0]=>[5,4,3,2,6,1]=>[5,4,3,2,6,1]
[3,3,2,2]=>[1,1,1,0,1,1,0,1,0,0,0,0]=>[3,5,6,4,2,1]=>[5,6,3,4,2,1]
[3,3,2,1,1]=>[1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[3,5,4,2,6,7,1]=>[5,3,4,2,6,7,1]
[3,3,1,1,1,1]=>[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[3,4,2,5,6,7,8,1]=>[3,4,2,5,6,7,8,1]
[3,2,2,2,1]=>[1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[1,5,6,4,3,7,2]=>[5,6,4,3,7,1,2]
[3,2,2,1,1,1]=>[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[1,5,4,3,6,7,8,2]=>[5,4,3,6,7,8,1,2]
[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]=>[1,4,3,5,6,7,8,9,2]=>[4,3,5,6,7,8,9,1,2]
[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]=>[1,2,4,5,6,7,8,9,10,3]=>[4,5,6,7,8,9,10,1,2,3]
[2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[4,5,6,3,2,1]=>[4,5,6,3,2,1]
[2,2,2,2,1,1]=>[1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[4,5,3,2,6,7,1]=>[4,5,3,2,6,7,1]
[2,2,2,1,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]=>[4,3,2,5,6,7,8,1]=>[4,3,2,5,6,7,8,1]
[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]=>[3,2,4,5,6,7,8,9,1]=>[3,2,4,5,6,7,8,9,1]
[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]=>[1,3,4,5,6,7,8,9,10,2]=>[3,4,5,6,7,8,9,10,1,2]
[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]=>[2,3,4,5,6,7,8,9,10,1]=>[2,3,4,5,6,7,8,9,10,1]
[11]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7,8,9,10,11]=>[1,2,3,4,5,6,7,8,9,10,11]
[10,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,6,7,8,9,11,10]=>[11,1,2,3,4,5,6,7,8,9,10]
[9,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,5,6,7,10,9,8]=>[10,9,1,2,3,4,5,6,7,8]
[9,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,2,3,4,5,6,7,8,10,11,9]=>[10,11,1,2,3,4,5,6,7,8,9]
[8,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[1,2,3,4,5,8,9,7,6]=>[8,9,7,1,2,3,4,5,6]
[7,4]=>[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[1,2,3,6,7,8,5,4]=>[6,7,8,5,1,2,3,4]
[7,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,4,5,9,8,7,6]=>[9,8,7,1,2,3,4,5,6]
[6,5]=>[1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[1,4,5,6,7,3,2]=>[4,5,6,7,3,1,2]
[6,4,1]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[1,2,5,6,7,4,8,3]=>[5,6,7,4,8,1,2,3]
[6,3,2]=>[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[1,2,3,6,8,7,5,4]=>[8,6,7,5,1,2,3,4]
[5,5,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[3,4,5,6,2,7,1]=>[3,4,5,6,2,7,1]
[5,4,2]=>[1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[1,4,5,7,6,3,2]=>[7,4,5,6,3,1,2]
[5,4,1,1]=>[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[1,4,5,6,3,7,8,2]=>[4,5,6,3,7,8,1,2]
[5,3,3]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,7,6,5,4,3]=>[7,6,5,4,1,2,3]
[5,3,2,1]=>[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[1,2,5,7,6,4,8,3]=>[7,5,6,4,8,1,2,3]
[5,3,1,1,1]=>[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[1,2,5,6,4,7,8,9,3]=>[5,6,4,7,8,9,1,2,3]
[5,2,2,2]=>[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[1,2,3,7,8,6,5,4]=>[7,8,6,5,1,2,3,4]
[5,2,2,1,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[1,2,3,7,6,5,8,9,4]=>[7,6,5,8,9,1,2,3,4]
[4,4,3]=>[1,1,1,0,1,1,1,0,0,0,0,0]=>[3,6,5,4,2,1]=>[6,5,3,4,2,1]
[4,4,2,1]=>[1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>[3,4,6,5,2,7,1]=>[6,3,4,5,2,7,1]
[4,4,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[3,4,5,2,6,7,8,1]=>[3,4,5,2,6,7,8,1]
[4,3,3,1]=>[1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>[1,6,5,4,3,7,2]=>[6,5,4,3,7,1,2]
[4,3,2,2]=>[1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>[1,4,6,7,5,3,2]=>[6,7,4,5,3,1,2]
[4,3,2,1,1]=>[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[1,4,6,5,3,7,8,2]=>[6,4,5,3,7,8,1,2]
[4,3,1,1,1,1]=>[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[1,4,5,3,6,7,8,9,2]=>[4,5,3,6,7,8,9,1,2]
[4,2,2,2,1]=>[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[1,2,6,7,5,4,8,3]=>[6,7,5,4,8,1,2,3]
[3,3,3,2]=>[1,1,1,1,1,0,0,1,0,0,0,0]=>[5,4,6,3,2,1]=>[5,4,6,3,2,1]
[3,3,3,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[5,4,3,2,6,7,1]=>[5,4,3,2,6,7,1]
[3,3,2,2,1]=>[1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[3,5,6,4,2,7,1]=>[5,6,3,4,2,7,1]
[3,3,2,1,1,1]=>[1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]=>[3,5,4,2,6,7,8,1]=>[5,3,4,2,6,7,8,1]
[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]=>[3,4,2,5,6,7,8,9,1]=>[3,4,2,5,6,7,8,9,1]
[3,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,5,6,7,4,3,2]=>[5,6,7,4,3,1,2]
[3,2,1,1,1,1,1,1]=>[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[1,4,3,5,6,7,8,9,10,2]=>[4,3,5,6,7,8,9,10,1,2]
[3,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[1,2,4,5,6,7,8,9,10,11,3]=>[4,5,6,7,8,9,10,11,1,2,3]
[2,2,2,2,2,1]=>[1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[4,5,6,3,2,7,1]=>[4,5,6,3,2,7,1]
[2,2,2,2,1,1,1]=>[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]=>[4,5,3,2,6,7,8,1]=>[4,5,3,2,6,7,8,1]
[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]=>[4,3,2,5,6,7,8,9,1]=>[4,3,2,5,6,7,8,9,1]
[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]=>[3,2,4,5,6,7,8,9,10,1]=>[3,2,4,5,6,7,8,9,10,1]
[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]=>[1,3,4,5,6,7,8,9,10,11,2]=>[3,4,5,6,7,8,9,10,11,1,2]
[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]=>[2,3,4,5,6,7,8,9,10,11,1]=>[2,3,4,5,6,7,8,9,10,11,1]
[10,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,5,6,7,8,11,10,9]=>[11,10,1,2,3,4,5,6,7,8,9]
[10,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[1,2,3,4,5,6,7,8,9,11,12,10]=>[11,12,1,2,3,4,5,6,7,8,9,10]
[8,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,4,5,6,10,9,8,7]=>[10,9,8,1,2,3,4,5,6,7]
[7,5]=>[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[1,2,5,6,7,8,4,3]=>[5,6,7,8,4,1,2,3]
[6,6]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[3,4,5,6,7,2,1]=>[3,4,5,6,7,2,1]
[6,5,1]=>[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[1,4,5,6,7,3,8,2]=>[4,5,6,7,3,8,1,2]
[6,4,2]=>[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[1,2,5,6,8,7,4,3]=>[8,5,6,7,4,1,2,3]
[6,4,1,1]=>[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[1,2,5,6,7,4,8,9,3]=>[5,6,7,4,8,9,1,2,3]
[6,3,3]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,3,8,7,6,5,4]=>[8,7,6,5,1,2,3,4]
[5,5,2]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[3,4,5,7,6,2,1]=>[7,3,4,5,6,2,1]
[5,5,1,1]=>[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>[3,4,5,6,2,7,8,1]=>[3,4,5,6,2,7,8,1]
[5,4,3]=>[1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[1,4,7,6,5,3,2]=>[7,6,4,5,3,1,2]
[5,4,2,1]=>[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>[1,4,5,7,6,3,8,2]=>[7,4,5,6,3,8,1,2]
[5,4,1,1,1]=>[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[1,4,5,6,3,7,8,9,2]=>[4,5,6,3,7,8,9,1,2]
[5,3,3,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>[1,2,7,6,5,4,8,3]=>[7,6,5,4,8,1,2,3]
[5,3,2,2]=>[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>[1,2,5,7,8,6,4,3]=>[7,8,5,6,4,1,2,3]
[5,3,2,1,1]=>[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[1,2,5,7,6,4,8,9,3]=>[7,5,6,4,8,9,1,2,3]
[5,2,2,2,1]=>[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[1,2,3,7,8,6,5,9,4]=>[7,8,6,5,9,1,2,3,4]
[4,4,4]=>[1,1,1,1,1,0,1,0,0,0,0,0]=>[5,6,4,3,2,1]=>[5,6,4,3,2,1]
[4,4,3,1]=>[1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>[3,6,5,4,2,7,1]=>[6,5,3,4,2,7,1]
[4,4,2,2]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[3,4,6,7,5,2,1]=>[6,7,3,4,5,2,1]
[4,4,2,1,1]=>[1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]=>[3,4,6,5,2,7,8,1]=>[6,3,4,5,2,7,8,1]
[4,4,1,1,1,1]=>[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[3,4,5,2,6,7,8,9,1]=>[3,4,5,2,6,7,8,9,1]
[4,3,3,2]=>[1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[1,6,5,7,4,3,2]=>[6,5,7,4,3,1,2]
[4,3,3,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[1,6,5,4,3,7,8,2]=>[6,5,4,3,7,8,1,2]
[4,3,2,2,1]=>[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[1,4,6,7,5,3,8,2]=>[6,7,4,5,3,8,1,2]
[4,2,2,2,2]=>[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[1,2,6,7,8,5,4,3]=>[6,7,8,5,4,1,2,3]
[3,3,3,3]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]
[3,3,3,2,1]=>[1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[5,4,6,3,2,7,1]=>[5,4,6,3,2,7,1]
[3,3,3,1,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]=>[5,4,3,2,6,7,8,1]=>[5,4,3,2,6,7,8,1]
[3,3,2,2,2]=>[1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[3,5,6,7,4,2,1]=>[5,6,7,3,4,2,1]
[3,3,2,2,1,1]=>[1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]=>[3,5,6,4,2,7,8,1]=>[5,6,3,4,2,7,8,1]
[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]=>[3,4,2,5,6,7,8,9,10,1]=>[3,4,2,5,6,7,8,9,10,1]
[3,2,2,2,2,1]=>[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[1,5,6,7,4,3,8,2]=>[5,6,7,4,3,8,1,2]
[2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[4,5,6,7,3,2,1]=>[4,5,6,7,3,2,1]
[2,2,2,2,2,1,1]=>[1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]=>[4,5,6,3,2,7,8,1]=>[4,5,6,3,2,7,8,1]
[2,2,2,1,1,1,1,1,1]=>[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[4,3,2,5,6,7,8,9,10,1]=>[4,3,2,5,6,7,8,9,10,1]
[2,2,1,1,1,1,1,1,1,1]=>[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[3,2,4,5,6,7,8,9,10,11,1]=>[3,2,4,5,6,7,8,9,10,11,1]
[2,1,1,1,1,1,1,1,1,1,1]=>[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[1,3,4,5,6,7,8,9,10,11,12,2]=>[3,4,5,6,7,8,9,10,11,12,1,2]
[7,6]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[1,4,5,6,7,8,3,2]=>[4,5,6,7,8,3,1,2]
[7,3,3]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,3,4,9,8,7,6,5]=>[9,8,7,6,1,2,3,4,5]
[6,6,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[3,4,5,6,7,2,8,1]=>[3,4,5,6,7,2,8,1]
[6,4,3]=>[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[1,2,5,8,7,6,4,3]=>[8,7,5,6,4,1,2,3]
[5,5,3]=>[1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>[3,4,7,6,5,2,1]=>[7,6,3,4,5,2,1]
[5,5,2,1]=>[1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]=>[3,4,5,7,6,2,8,1]=>[7,3,4,5,6,2,8,1]
[5,4,4]=>[1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[1,6,7,5,4,3,2]=>[6,7,5,4,3,1,2]
[5,4,3,1]=>[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>[1,4,7,6,5,3,8,2]=>[7,6,4,5,3,8,1,2]
[5,4,2,2]=>[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[1,4,5,7,8,6,3,2]=>[7,8,4,5,6,3,1,2]
[5,4,2,1,1]=>[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]=>[1,4,5,7,6,3,8,9,2]=>[7,4,5,6,3,8,9,1,2]
[5,3,3,2]=>[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[1,2,7,6,8,5,4,3]=>[7,6,8,5,4,1,2,3]
[5,3,3,1,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[1,2,7,6,5,4,8,9,3]=>[7,6,5,4,8,9,1,2,3]
[5,3,2,2,1]=>[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[1,2,5,7,8,6,4,9,3]=>[7,8,5,6,4,9,1,2,3]
[4,4,4,1]=>[1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[5,6,4,3,2,7,1]=>[5,6,4,3,2,7,1]
[4,4,3,2]=>[1,1,1,0,1,1,1,0,0,1,0,0,0,0]=>[3,6,5,7,4,2,1]=>[6,5,7,3,4,2,1]
[4,4,3,1,1]=>[1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]=>[3,6,5,4,2,7,8,1]=>[6,5,3,4,2,7,8,1]
[4,4,2,2,1]=>[1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]=>[3,4,6,7,5,2,8,1]=>[6,7,3,4,5,2,8,1]
[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]=>[3,4,5,2,6,7,8,9,10,1]=>[3,4,5,2,6,7,8,9,10,1]
[4,3,3,3]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,7,6,5,4,3,2]=>[7,6,5,4,3,1,2]
[4,3,3,2,1]=>[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[1,6,5,7,4,3,8,2]=>[6,5,7,4,3,8,1,2]
[4,3,2,2,2]=>[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[1,4,6,7,8,5,3,2]=>[6,7,8,4,5,3,1,2]
[3,3,3,3,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[6,5,4,3,2,7,1]=>[6,5,4,3,2,7,1]
[3,3,3,2,2]=>[1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[5,4,6,7,3,2,1]=>[5,4,6,7,3,2,1]
[3,3,3,2,1,1]=>[1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]=>[5,4,6,3,2,7,8,1]=>[5,4,6,3,2,7,8,1]
[3,3,3,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,0]=>[5,4,3,2,6,7,8,9,1]=>[5,4,3,2,6,7,8,9,1]
[3,3,2,2,2,1]=>[1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]=>[3,5,6,7,4,2,8,1]=>[5,6,7,3,4,2,8,1]
[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]=>[3,4,2,5,6,7,8,9,10,11,1]=>[3,4,2,5,6,7,8,9,10,11,1]
[3,2,2,2,2,2]=>[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[1,5,6,7,8,4,3,2]=>[5,6,7,8,4,3,1,2]
[2,2,2,2,2,2,1]=>[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]=>[4,5,6,7,3,2,8,1]=>[4,5,6,7,3,2,8,1]
[8,3,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,3,4,5,10,9,8,7,6]=>[10,9,8,7,1,2,3,4,5,6]
[7,7]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[3,4,5,6,7,8,2,1]=>[3,4,5,6,7,8,2,1]
[7,6,1]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[1,4,5,6,7,8,3,9,2]=>[4,5,6,7,8,3,9,1,2]
[6,6,2]=>[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[3,4,5,6,8,7,2,1]=>[8,3,4,5,6,7,2,1]
[6,6,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>[3,4,5,6,7,2,8,9,1]=>[3,4,5,6,7,2,8,9,1]
[6,5,3]=>[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>[1,4,5,8,7,6,3,2]=>[8,7,4,5,6,3,1,2]
[6,4,4]=>[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[1,2,7,8,6,5,4,3]=>[7,8,6,5,4,1,2,3]
[5,5,4]=>[1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>[3,6,7,5,4,2,1]=>[6,7,5,3,4,2,1]
[5,5,3,1]=>[1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]=>[3,4,7,6,5,2,8,1]=>[7,6,3,4,5,2,8,1]
[5,5,2,2]=>[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]=>[3,4,5,7,8,6,2,1]=>[7,8,3,4,5,6,2,1]
[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]=>[3,4,5,6,2,7,8,9,10,1]=>[3,4,5,6,2,7,8,9,10,1]
[5,4,4,1]=>[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[1,6,7,5,4,3,8,2]=>[6,7,5,4,3,8,1,2]
[5,4,3,2]=>[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]=>[1,4,7,6,8,5,3,2]=>[7,6,8,4,5,3,1,2]
[5,4,3,1,1]=>[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]=>[1,4,7,6,5,3,8,9,2]=>[7,6,4,5,3,8,9,1,2]
[5,4,2,2,1]=>[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]=>[1,4,5,7,8,6,3,9,2]=>[7,8,4,5,6,3,9,1,2]
[5,3,3,3]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,8,7,6,5,4,3]=>[8,7,6,5,4,1,2,3]
[5,3,3,2,1]=>[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[1,2,7,6,8,5,4,9,3]=>[7,6,8,5,4,9,1,2,3]
[4,4,4,2]=>[1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>[5,6,4,7,3,2,1]=>[5,6,4,7,3,2,1]
[4,4,3,3]=>[1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[3,7,6,5,4,2,1]=>[7,6,5,3,4,2,1]
[4,4,3,2,1]=>[1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]=>[3,6,5,7,4,2,8,1]=>[6,5,7,3,4,2,8,1]
[4,4,2,2,2]=>[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]=>[3,4,6,7,8,5,2,1]=>[6,7,8,3,4,5,2,1]
[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]=>[3,4,5,2,6,7,8,9,10,11,1]=>[3,4,5,2,6,7,8,9,10,11,1]
[4,3,3,3,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[1,7,6,5,4,3,8,2]=>[7,6,5,4,3,8,1,2]
[4,3,3,2,2]=>[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[1,6,5,7,8,4,3,2]=>[6,5,7,8,4,3,1,2]
[3,3,3,3,2]=>[1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[6,5,4,7,3,2,1]=>[6,5,4,7,3,2,1]
[3,3,3,3,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]=>[6,5,4,3,2,7,8,1]=>[6,5,4,3,2,7,8,1]
[3,3,3,2,2,1]=>[1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]=>[5,4,6,7,3,2,8,1]=>[5,4,6,7,3,2,8,1]
[3,3,3,1,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[5,4,3,2,6,7,8,9,10,1]=>[5,4,3,2,6,7,8,9,10,1]
[3,3,2,2,2,2]=>[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]=>[3,5,6,7,8,4,2,1]=>[5,6,7,8,3,4,2,1]
[2,2,2,2,2,2,2]=>[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[4,5,6,7,8,3,2,1]=>[4,5,6,7,8,3,2,1]
[8,7]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[1,4,5,6,7,8,9,3,2]=>[4,5,6,7,8,9,3,1,2]
[7,7,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[3,4,5,6,7,8,2,9,1]=>[3,4,5,6,7,8,2,9,1]
[6,6,3]=>[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]=>[3,4,5,8,7,6,2,1]=>[8,7,3,4,5,6,2,1]
[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]=>[3,4,5,6,7,2,8,9,10,1]=>[3,4,5,6,7,2,8,9,10,1]
[6,5,4]=>[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>[1,4,7,8,6,5,3,2]=>[7,8,6,4,5,3,1,2]
[6,3,3,3]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,3,9,8,7,6,5,4]=>[9,8,7,6,5,1,2,3,4]
[5,5,5]=>[1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[5,6,7,4,3,2,1]=>[5,6,7,4,3,2,1]
[5,5,4,1]=>[1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]=>[3,6,7,5,4,2,8,1]=>[6,7,5,3,4,2,8,1]
[5,5,3,2]=>[1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]=>[3,4,7,6,8,5,2,1]=>[7,6,8,3,4,5,2,1]
[5,4,4,2]=>[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>[1,6,7,5,8,4,3,2]=>[6,7,5,8,4,3,1,2]
[5,4,3,3]=>[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[1,4,8,7,6,5,3,2]=>[8,7,6,4,5,3,1,2]
[5,4,3,2,1]=>[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]=>[1,4,7,6,8,5,3,9,2]=>[7,6,8,4,5,3,9,1,2]
[4,4,4,3]=>[1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[5,7,6,4,3,2,1]=>[7,5,6,4,3,2,1]
[4,4,4,2,1]=>[1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]=>[5,6,4,7,3,2,8,1]=>[5,6,4,7,3,2,8,1]
[4,4,3,3,1]=>[1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]=>[3,7,6,5,4,2,8,1]=>[7,6,5,3,4,2,8,1]
[4,4,3,2,2]=>[1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]=>[3,6,5,7,8,4,2,1]=>[6,5,7,8,3,4,2,1]
[4,3,3,3,2]=>[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[1,7,6,5,8,4,3,2]=>[7,6,5,8,4,3,1,2]
[3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[6,7,5,4,3,2,1]=>[6,7,5,4,3,2,1]
[3,3,3,3,2,1]=>[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]=>[6,5,4,7,3,2,8,1]=>[6,5,4,7,3,2,8,1]
[3,3,3,3,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]=>[6,5,4,3,2,7,8,9,1]=>[6,5,4,3,2,7,8,9,1]
[3,3,3,2,2,2]=>[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]=>[5,4,6,7,8,3,2,1]=>[5,4,6,7,8,3,2,1]
[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]=>[1,5,6,7,8,9,4,3,2]=>[5,6,7,8,9,4,3,1,2]
[8,8]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[3,4,5,6,7,8,9,2,1]=>[3,4,5,6,7,8,9,2,1]
[7,7,1,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]=>[3,4,5,6,7,8,2,9,10,1]=>[3,4,5,6,7,8,2,9,10,1]
[7,3,3,3]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,3,4,10,9,8,7,6,5]=>[10,9,8,7,6,1,2,3,4,5]
[6,6,4]=>[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]=>[3,4,7,8,6,5,2,1]=>[7,8,6,3,4,5,2,1]
[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]=>[3,4,5,6,7,2,8,9,10,11,1]=>[3,4,5,6,7,2,8,9,10,11,1]
[6,5,5]=>[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[1,6,7,8,5,4,3,2]=>[6,7,8,5,4,3,1,2]
[5,5,5,1]=>[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]=>[5,6,7,4,3,2,8,1]=>[5,6,7,4,3,2,8,1]
[5,5,4,2]=>[1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]=>[3,6,7,5,8,4,2,1]=>[6,7,5,8,3,4,2,1]
[5,5,3,3]=>[1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]=>[3,4,8,7,6,5,2,1]=>[8,7,6,3,4,5,2,1]
[5,4,4,3]=>[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[1,6,8,7,5,4,3,2]=>[8,6,7,5,4,3,1,2]
[4,4,4,4]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[7,6,5,4,3,2,1]=>[7,6,5,4,3,2,1]
[4,4,4,3,1]=>[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]=>[5,7,6,4,3,2,8,1]=>[7,5,6,4,3,2,8,1]
[4,4,4,2,2]=>[1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]=>[5,6,4,7,8,3,2,1]=>[5,6,4,7,8,3,2,1]
[4,4,3,3,2]=>[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]=>[3,7,6,5,8,4,2,1]=>[7,6,5,8,3,4,2,1]
[4,3,3,3,3]=>[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[1,7,8,6,5,4,3,2]=>[7,8,6,5,4,3,1,2]
[4,3,3,2,2,2]=>[1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]=>[1,6,5,7,8,9,4,3,2]=>[6,5,7,8,9,4,3,1,2]
[3,3,3,3,3,1]=>[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]=>[6,7,5,4,3,2,8,1]=>[6,7,5,4,3,2,8,1]
[3,3,3,3,2,2]=>[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]=>[6,5,4,7,8,3,2,1]=>[6,5,4,7,8,3,2,1]
[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]=>[6,5,4,3,2,7,8,9,10,1]=>[6,5,4,3,2,7,8,9,10,1]
[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]=>[3,5,6,7,8,9,4,2,1]=>[5,6,7,8,9,3,4,2,1]
[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]=>[4,5,6,7,8,9,3,2,1]=>[4,5,6,7,8,9,3,2,1]
[9,8]=>[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[1,4,5,6,7,8,9,10,3,2]=>[4,5,6,7,8,9,10,3,1,2]
[8,8,1]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[3,4,5,6,7,8,9,2,10,1]=>[3,4,5,6,7,8,9,2,10,1]
[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]=>[3,4,5,6,7,8,2,9,10,11,1]=>[3,4,5,6,7,8,2,9,10,11,1]
[6,6,5]=>[1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]=>[3,6,7,8,5,4,2,1]=>[6,7,8,5,3,4,2,1]
[5,5,5,2]=>[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]=>[5,6,7,4,8,3,2,1]=>[5,6,7,4,8,3,2,1]
[5,5,4,3]=>[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]=>[3,6,8,7,5,4,2,1]=>[8,6,7,5,3,4,2,1]
[5,4,4,4]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,8,7,6,5,4,3,2]=>[8,7,6,5,4,3,1,2]
[5,3,3,3,3]=>[1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[1,2,8,9,7,6,5,4,3]=>[8,9,7,6,5,4,1,2,3]
[4,4,4,4,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]=>[7,6,5,4,3,2,8,1]=>[7,6,5,4,3,2,8,1]
[4,4,4,3,2]=>[1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]=>[5,7,6,4,8,3,2,1]=>[7,5,6,4,8,3,2,1]
[4,4,3,3,3]=>[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]=>[3,7,8,6,5,4,2,1]=>[7,8,6,5,3,4,2,1]
[3,3,3,3,3,2]=>[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]=>[6,7,5,4,8,3,2,1]=>[6,7,5,4,8,3,2,1]
[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]=>[5,4,6,7,8,9,3,2,1]=>[5,4,6,7,8,9,3,2,1]
[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]=>[1,5,6,7,8,9,10,4,3,2]=>[5,6,7,8,9,10,4,3,1,2]
[]=>[]=>[]=>[]
[6,5,4,2,1]=>[1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]=>[1,4,7,8,6,9,5,3,10,2]=>[7,8,6,9,4,5,3,10,1,2]
[6,5,4,3,3,2,1]=>[1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0]=>[1,4,7,9,10,8,6,11,5,3,12,2]=>[9,10,7,8,6,11,4,5,3,12,1,2]
[4,4,4,4,3,1]=>[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0]=>[7,6,8,5,4,3,2,9,1]=>[7,6,8,5,4,3,2,9,1]
[5,4,4,4,3]=>[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]=>[1,8,7,9,6,5,4,3,2]=>[8,7,9,6,5,4,3,1,2]
[4,4,4,4,2]=>[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]=>[7,6,5,4,8,3,2,1]=>[7,6,5,4,8,3,2,1]
[6,5,5,4]=>[1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]=>[1,6,9,8,7,5,4,3,2]=>[9,8,6,7,5,4,3,1,2]
[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]=>[4,5,6,7,8,9,10,3,2,1]=>[4,5,6,7,8,9,10,3,2,1]
[3,3,3,3,3,3]=>[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]=>[6,7,8,5,4,3,2,1]=>[6,7,8,5,4,3,2,1]
[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]=>[6,7,8,9,10,5,4,3,2,1]=>[6,7,8,9,10,5,4,3,2,1]
[3,3,3,3,3,3,1]=>[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]=>[6,7,8,5,4,3,2,9,1]=>[6,7,8,5,4,3,2,9,1]
[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]=>[6,7,8,9,5,4,3,2,1]=>[6,7,8,9,5,4,3,2,1]
[4,3,3,3,3,3]=>[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]=>[1,7,8,9,6,5,4,3,2]=>[7,8,9,6,5,4,3,1,2]
[4,4,4,4,4]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]
[4,4,4,3,3,3]=>[1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0]=>[5,7,8,9,6,4,3,2,1]=>[7,8,9,5,6,4,3,2,1]
[4,4,4,4,4,4]=>[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]=>[8,9,7,6,5,4,3,2,1]=>[8,9,7,6,5,4,3,2,1]
[5,5,5,5]=>[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]=>[7,8,6,5,4,3,2,1]=>[7,8,6,5,4,3,2,1]
[6,6,6,6]=>[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]=>[7,8,9,6,5,4,3,2,1]=>[7,8,9,6,5,4,3,2,1]
[6,6,6]=>[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[5,6,7,8,4,3,2,1]=>[5,6,7,8,4,3,2,1]
[7,6,6]=>[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[1,6,7,8,9,5,4,3,2]=>[6,7,8,9,5,4,3,1,2]
[10,10]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[3,4,5,6,7,8,9,10,11,2,1]=>[3,4,5,6,7,8,9,10,11,2,1]
[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]=>[9,10,8,7,6,5,4,3,2,1]=>[9,10,8,7,6,5,4,3,2,1]
[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]=>[10,9,8,7,6,5,4,3,2,1]=>[10,9,8,7,6,5,4,3,2,1]
[7,7,7,7]=>[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]=>[7,8,9,10,6,5,4,3,2,1]=>[7,8,9,10,6,5,4,3,2,1]
[7,7,7]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]=>[5,6,7,8,9,4,3,2,1]=>[5,6,7,8,9,4,3,2,1]
[5,5,5,5,5]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[9,8,7,6,5,4,3,2,1]=>[9,8,7,6,5,4,3,2,1]
[9,9]=>[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[3,4,5,6,7,8,9,10,2,1]=>[3,4,5,6,7,8,9,10,2,1]
[8,8,8]=>[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]=>[5,6,7,8,9,10,4,3,2,1]=>[5,6,7,8,9,10,4,3,2,1]
[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]=>[8,9,10,7,6,5,4,3,2,1]=>[8,9,10,7,6,5,4,3,2,1]
[6,5,5,5]=>[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]=>[1,8,9,7,6,5,4,3,2]=>[8,9,7,6,5,4,3,1,2]
[5,4,4,4,4]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,9,8,7,6,5,4,3,2]=>[9,8,7,6,5,4,3,1,2]
[4,4,4,4,4,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]=>[8,7,6,5,4,3,2,9,1]=>[8,7,6,5,4,3,2,9,1]
[5,5,5,5,1]=>[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]=>[7,8,6,5,4,3,2,9,1]=>[7,8,6,5,4,3,2,9,1]
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 312-avoiding permutation
Description
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
See Mp00067Foata bijection.
searching the database
Sorry, this map was not found in the database.