Identifier
Mp00039:
Integer compositions
—complement⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Images
=>
Cc0005;cc-rep-2
[1]=>[1]=>[1,0]=>[1]
[1,1]=>[2]=>[1,1,0,0]=>[2,1]
[2]=>[1,1]=>[1,0,1,0]=>[1,2]
[1,1,1]=>[3]=>[1,1,1,0,0,0]=>[3,1,2]
[1,2]=>[2,1]=>[1,1,0,0,1,0]=>[2,1,3]
[2,1]=>[1,2]=>[1,0,1,1,0,0]=>[1,3,2]
[3]=>[1,1,1]=>[1,0,1,0,1,0]=>[1,2,3]
[1,1,1,1]=>[4]=>[1,1,1,1,0,0,0,0]=>[4,1,2,3]
[1,1,2]=>[3,1]=>[1,1,1,0,0,0,1,0]=>[3,1,2,4]
[1,2,1]=>[2,2]=>[1,1,0,0,1,1,0,0]=>[2,1,4,3]
[1,3]=>[2,1,1]=>[1,1,0,0,1,0,1,0]=>[2,1,3,4]
[2,1,1]=>[1,3]=>[1,0,1,1,1,0,0,0]=>[1,4,2,3]
[2,2]=>[1,2,1]=>[1,0,1,1,0,0,1,0]=>[1,3,2,4]
[3,1]=>[1,1,2]=>[1,0,1,0,1,1,0,0]=>[1,2,4,3]
[4]=>[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[1,2,3,4]
[1,1,1,1,1]=>[5]=>[1,1,1,1,1,0,0,0,0,0]=>[5,1,2,3,4]
[1,1,1,2]=>[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[4,1,2,3,5]
[1,1,2,1]=>[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[3,1,2,5,4]
[1,1,3]=>[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[3,1,2,4,5]
[1,2,1,1]=>[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[2,1,5,3,4]
[1,2,2]=>[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,5]
[1,3,1]=>[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[2,1,3,5,4]
[1,4]=>[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[2,1,3,4,5]
[2,1,1,1]=>[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[1,5,2,3,4]
[2,1,2]=>[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[1,4,2,3,5]
[2,2,1]=>[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4]
[2,3]=>[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[1,3,2,4,5]
[3,1,1]=>[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[1,2,5,3,4]
[3,2]=>[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[1,2,4,3,5]
[4,1]=>[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[1,2,3,5,4]
[5]=>[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5]
[1,1,1,1,1,1]=>[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[6,1,2,3,4,5]
[1,1,1,1,2]=>[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[5,1,2,3,4,6]
[1,1,1,2,1]=>[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[4,1,2,3,6,5]
[1,1,1,3]=>[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[4,1,2,3,5,6]
[1,1,2,1,1]=>[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[3,1,2,6,4,5]
[1,1,2,2]=>[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[3,1,2,5,4,6]
[1,1,3,1]=>[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[3,1,2,4,6,5]
[1,1,4]=>[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[3,1,2,4,5,6]
[1,2,1,1,1]=>[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,6,3,4,5]
[1,2,1,2]=>[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[2,1,5,3,4,6]
[1,2,2,1]=>[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5]
[1,2,3]=>[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[2,1,4,3,5,6]
[1,3,1,1]=>[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[2,1,3,6,4,5]
[1,3,2]=>[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[2,1,3,5,4,6]
[1,4,1]=>[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,6,5]
[1,5]=>[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6]
[2,1,1,1,1]=>[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[1,6,2,3,4,5]
[2,1,1,2]=>[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[1,5,2,3,4,6]
[2,1,2,1]=>[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[1,4,2,3,6,5]
[2,1,3]=>[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[1,4,2,3,5,6]
[2,2,1,1]=>[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[1,3,2,6,4,5]
[2,2,2]=>[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,6]
[2,3,1]=>[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[1,3,2,4,6,5]
[2,4]=>[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,2,4,5,6]
[3,1,1,1]=>[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,6,3,4,5]
[3,1,2]=>[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,5,3,4,6]
[3,2,1]=>[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,6,5]
[3,3]=>[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,5,6]
[4,1,1]=>[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,6,4,5]
[4,2]=>[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,5,4,6]
[5,1]=>[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,6,5]
[6]=>[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6]
[1,1,1,1,1,1,1]=>[7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[7,1,2,3,4,5,6]
[1,1,1,1,1,2]=>[6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[6,1,2,3,4,5,7]
[1,1,1,1,2,1]=>[5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[5,1,2,3,4,7,6]
[1,1,1,1,3]=>[5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[5,1,2,3,4,6,7]
[1,1,1,2,1,1]=>[4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[4,1,2,3,7,5,6]
[1,1,1,2,2]=>[4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[4,1,2,3,6,5,7]
[1,1,1,3,1]=>[4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[4,1,2,3,5,7,6]
[1,1,1,4]=>[4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[4,1,2,3,5,6,7]
[1,1,2,1,1,1]=>[3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[3,1,2,7,4,5,6]
[1,1,2,1,2]=>[3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[3,1,2,6,4,5,7]
[1,1,2,2,1]=>[3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[3,1,2,5,4,7,6]
[1,1,2,3]=>[3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[3,1,2,5,4,6,7]
[1,1,3,1,1]=>[3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[3,1,2,4,7,5,6]
[1,1,3,2]=>[3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[3,1,2,4,6,5,7]
[1,1,4,1]=>[3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[3,1,2,4,5,7,6]
[1,1,5]=>[3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[3,1,2,4,5,6,7]
[1,2,1,1,1,1]=>[2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,7,3,4,5,6]
[1,2,1,1,2]=>[2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[2,1,6,3,4,5,7]
[1,2,1,2,1]=>[2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[2,1,5,3,4,7,6]
[1,2,1,3]=>[2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[2,1,5,3,4,6,7]
[1,2,2,1,1]=>[2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,4,3,7,5,6]
[1,2,2,2]=>[2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5,7]
[1,2,3,1]=>[2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[2,1,4,3,5,7,6]
[1,2,4]=>[2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,4,3,5,6,7]
[1,3,1,1,1]=>[2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[2,1,3,7,4,5,6]
[1,3,1,2]=>[2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[2,1,3,6,4,5,7]
[1,3,2,1]=>[2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[2,1,3,5,4,7,6]
[1,3,3]=>[2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[2,1,3,5,4,6,7]
[1,4,1,1]=>[2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[2,1,3,4,7,5,6]
[1,4,2]=>[2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[2,1,3,4,6,5,7]
[1,5,1]=>[2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,7,6]
[1,6]=>[2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6,7]
[2,1,1,1,1,1]=>[1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,7,2,3,4,5,6]
[2,1,1,1,2]=>[1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,6,2,3,4,5,7]
[2,1,1,2,1]=>[1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[1,5,2,3,4,7,6]
[2,1,1,3]=>[1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[1,5,2,3,4,6,7]
[2,1,2,1,1]=>[1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,4,2,3,7,5,6]
[2,1,2,2]=>[1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,4,2,3,6,5,7]
[2,1,3,1]=>[1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[1,4,2,3,5,7,6]
[2,1,4]=>[1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,4,2,3,5,6,7]
[2,2,1,1,1]=>[1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,3,2,7,4,5,6]
[2,2,1,2]=>[1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[1,3,2,6,4,5,7]
[2,2,2,1]=>[1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4,7,6]
[2,2,3]=>[1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,3,2,5,4,6,7]
[2,3,1,1]=>[1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[1,3,2,4,7,5,6]
[2,3,2]=>[1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,6,5,7]
[2,4,1]=>[1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,3,2,4,5,7,6]
[2,5]=>[1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,3,2,4,5,6,7]
[3,1,1,1,1]=>[1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,7,3,4,5,6]
[3,1,1,2]=>[1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,2,6,3,4,5,7]
[3,1,2,1]=>[1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,2,5,3,4,7,6]
[3,1,3]=>[1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,2,5,3,4,6,7]
[3,2,1,1]=>[1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,2,4,3,7,5,6]
[3,2,2]=>[1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,4,3,6,5,7]
[3,3,1]=>[1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,2,4,3,5,7,6]
[3,4]=>[1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,4,3,5,6,7]
[4,1,1,1]=>[1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,7,4,5,6]
[4,1,2]=>[1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,3,6,4,5,7]
[4,2,1]=>[1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,3,5,4,7,6]
[4,3]=>[1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,3,5,4,6,7]
[5,1,1]=>[1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,7,5,6]
[5,2]=>[1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,4,6,5,7]
[6,1]=>[1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,4,5,7,6]
[7]=>[1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7]
[1,1,1,1,1,1,1,1]=>[8]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[8,1,2,3,4,5,6,7]
[1,1,1,1,1,1,2]=>[7,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[7,1,2,3,4,5,6,8]
[1,1,1,1,1,2,1]=>[6,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[6,1,2,3,4,5,8,7]
[1,1,1,1,1,3]=>[6,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[6,1,2,3,4,5,7,8]
[1,1,1,1,2,1,1]=>[5,3]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>[5,1,2,3,4,8,6,7]
[1,1,1,1,2,2]=>[5,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]=>[5,1,2,3,4,7,6,8]
[1,1,1,1,3,1]=>[5,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]=>[5,1,2,3,4,6,8,7]
[1,1,1,1,4]=>[5,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[5,1,2,3,4,6,7,8]
[1,1,1,2,1,1,1]=>[4,4]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[4,1,2,3,8,5,6,7]
[1,1,1,2,1,2]=>[4,3,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]=>[4,1,2,3,7,5,6,8]
[1,1,1,2,2,1]=>[4,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]=>[4,1,2,3,6,5,8,7]
[1,1,1,2,3]=>[4,2,1,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]=>[4,1,2,3,6,5,7,8]
[1,1,1,3,1,1]=>[4,1,3]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]=>[4,1,2,3,5,8,6,7]
[1,1,1,3,2]=>[4,1,2,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]=>[4,1,2,3,5,7,6,8]
[1,1,1,4,1]=>[4,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]=>[4,1,2,3,5,6,8,7]
[1,1,1,5]=>[4,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[4,1,2,3,5,6,7,8]
[1,1,2,1,1,1,1]=>[3,5]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[3,1,2,8,4,5,6,7]
[1,1,2,1,1,2]=>[3,4,1]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[3,1,2,7,4,5,6,8]
[1,1,2,1,2,1]=>[3,3,2]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]=>[3,1,2,6,4,5,8,7]
[1,1,2,1,3]=>[3,3,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]=>[3,1,2,6,4,5,7,8]
[1,1,2,2,1,1]=>[3,2,3]=>[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]=>[3,1,2,5,4,8,6,7]
[1,1,2,2,2]=>[3,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[3,1,2,5,4,7,6,8]
[1,1,2,3,1]=>[3,2,1,2]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]=>[3,1,2,5,4,6,8,7]
[1,1,2,4]=>[3,2,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]=>[3,1,2,5,4,6,7,8]
[1,1,3,1,1,1]=>[3,1,4]=>[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]=>[3,1,2,4,8,5,6,7]
[1,1,3,1,2]=>[3,1,3,1]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]=>[3,1,2,4,7,5,6,8]
[1,1,3,2,1]=>[3,1,2,2]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]=>[3,1,2,4,6,5,8,7]
[1,1,3,3]=>[3,1,2,1,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]=>[3,1,2,4,6,5,7,8]
[1,1,4,1,1]=>[3,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[3,1,2,4,5,8,6,7]
[1,1,4,2]=>[3,1,1,2,1]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]=>[3,1,2,4,5,7,6,8]
[1,1,5,1]=>[3,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[3,1,2,4,5,6,8,7]
[1,1,6]=>[3,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[3,1,2,4,5,6,7,8]
[1,2,1,1,1,1,1]=>[2,6]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,1,8,3,4,5,6,7]
[1,2,1,1,1,2]=>[2,5,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[2,1,7,3,4,5,6,8]
[1,2,1,1,2,1]=>[2,4,2]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[2,1,6,3,4,5,8,7]
[1,2,1,1,3]=>[2,4,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[2,1,6,3,4,5,7,8]
[1,2,1,2,1,1]=>[2,3,3]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[2,1,5,3,4,8,6,7]
[1,2,1,2,2]=>[2,3,2,1]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[2,1,5,3,4,7,6,8]
[1,2,1,3,1]=>[2,3,1,2]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[2,1,5,3,4,6,8,7]
[1,2,1,4]=>[2,3,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[2,1,5,3,4,6,7,8]
[1,2,2,1,1,1]=>[2,2,4]=>[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,4,3,8,5,6,7]
[1,2,2,1,2]=>[2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[2,1,4,3,7,5,6,8]
[1,2,2,2,1]=>[2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5,8,7]
[1,2,2,3]=>[2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[2,1,4,3,6,5,7,8]
[1,2,3,1,1]=>[2,2,1,3]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[2,1,4,3,5,8,6,7]
[1,2,3,2]=>[2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[2,1,4,3,5,7,6,8]
[1,2,4,1]=>[2,2,1,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[2,1,4,3,5,6,8,7]
[1,2,5]=>[2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,4,3,5,6,7,8]
[1,3,1,1,1,1]=>[2,1,5]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,3,8,4,5,6,7]
[1,3,1,1,2]=>[2,1,4,1]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[2,1,3,7,4,5,6,8]
[1,3,1,2,1]=>[2,1,3,2]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[2,1,3,6,4,5,8,7]
[1,3,1,3]=>[2,1,3,1,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[2,1,3,6,4,5,7,8]
[1,3,2,1,1]=>[2,1,2,3]=>[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,3,5,4,8,6,7]
[1,3,2,2]=>[2,1,2,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,3,5,4,7,6,8]
[1,3,3,1]=>[2,1,2,1,2]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[2,1,3,5,4,6,8,7]
[1,3,4]=>[2,1,2,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,3,5,4,6,7,8]
[1,4,1,1,1]=>[2,1,1,4]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[2,1,3,4,8,5,6,7]
[1,4,1,2]=>[2,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[2,1,3,4,7,5,6,8]
[1,4,2,1]=>[2,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[2,1,3,4,6,5,8,7]
[1,4,3]=>[2,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[2,1,3,4,6,5,7,8]
[1,5,1,1]=>[2,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[2,1,3,4,5,8,6,7]
[1,5,2]=>[2,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[2,1,3,4,5,7,6,8]
[1,6,1]=>[2,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,6,8,7]
[1,7]=>[2,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6,7,8]
[2,1,1,1,1,1,1]=>[1,7]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,8,2,3,4,5,6,7]
[2,1,1,1,1,2]=>[1,6,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[1,7,2,3,4,5,6,8]
[2,1,1,1,2,1]=>[1,5,2]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[1,6,2,3,4,5,8,7]
[2,1,1,1,3]=>[1,5,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[1,6,2,3,4,5,7,8]
[2,1,1,2,1,1]=>[1,4,3]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[1,5,2,3,4,8,6,7]
[2,1,1,2,2]=>[1,4,2,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[1,5,2,3,4,7,6,8]
[2,1,1,3,1]=>[1,4,1,2]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[1,5,2,3,4,6,8,7]
[2,1,1,4]=>[1,4,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[1,5,2,3,4,6,7,8]
[2,1,2,1,1,1]=>[1,3,4]=>[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[1,4,2,3,8,5,6,7]
[2,1,2,1,2]=>[1,3,3,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[1,4,2,3,7,5,6,8]
[2,1,2,2,1]=>[1,3,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[1,4,2,3,6,5,8,7]
[2,1,2,3]=>[1,3,2,1,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[1,4,2,3,6,5,7,8]
[2,1,3,1,1]=>[1,3,1,3]=>[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[1,4,2,3,5,8,6,7]
[2,1,3,2]=>[1,3,1,2,1]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[1,4,2,3,5,7,6,8]
[2,1,4,1]=>[1,3,1,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[1,4,2,3,5,6,8,7]
[2,1,5]=>[1,3,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[1,4,2,3,5,6,7,8]
[2,2,1,1,1,1]=>[1,2,5]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,3,2,8,4,5,6,7]
[2,2,1,1,2]=>[1,2,4,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[1,3,2,7,4,5,6,8]
[2,2,1,2,1]=>[1,2,3,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[1,3,2,6,4,5,8,7]
[2,2,1,3]=>[1,2,3,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[1,3,2,6,4,5,7,8]
[2,2,2,1,1]=>[1,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[1,3,2,5,4,8,6,7]
[2,2,2,2]=>[1,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,7,6,8]
[2,2,3,1]=>[1,2,2,1,2]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[1,3,2,5,4,6,8,7]
[2,2,4]=>[1,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,2,5,4,6,7,8]
[2,3,1,1,1]=>[1,2,1,4]=>[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[1,3,2,4,8,5,6,7]
[2,3,1,2]=>[1,2,1,3,1]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[1,3,2,4,7,5,6,8]
[2,3,2,1]=>[1,2,1,2,2]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[1,3,2,4,6,5,8,7]
[2,3,3]=>[1,2,1,2,1,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[1,3,2,4,6,5,7,8]
[2,4,1,1]=>[1,2,1,1,3]=>[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[1,3,2,4,5,8,6,7]
[2,4,2]=>[1,2,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,5,7,6,8]
[2,5,1]=>[1,2,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[1,3,2,4,5,6,8,7]
[2,6]=>[1,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,3,2,4,5,6,7,8]
[3,1,1,1,1,1]=>[1,1,6]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,8,3,4,5,6,7]
[3,1,1,1,2]=>[1,1,5,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,2,7,3,4,5,6,8]
[3,1,1,2,1]=>[1,1,4,2]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[1,2,6,3,4,5,8,7]
[3,1,1,3]=>[1,1,4,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[1,2,6,3,4,5,7,8]
[3,1,2,1,1]=>[1,1,3,3]=>[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,2,5,3,4,8,6,7]
[3,1,2,2]=>[1,1,3,2,1]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,2,5,3,4,7,6,8]
[3,1,3,1]=>[1,1,3,1,2]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[1,2,5,3,4,6,8,7]
[3,1,4]=>[1,1,3,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,2,5,3,4,6,7,8]
[3,2,1,1,1]=>[1,1,2,4]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,2,4,3,8,5,6,7]
[3,2,1,2]=>[1,1,2,3,1]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[1,2,4,3,7,5,6,8]
[3,2,2,1]=>[1,1,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,6,5,8,7]
[3,2,3]=>[1,1,2,2,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,6,5,7,8]
[3,3,1,1]=>[1,1,2,1,3]=>[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[1,2,4,3,5,8,6,7]
[3,3,2]=>[1,1,2,1,2,1]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,2,4,3,5,7,6,8]
[3,4,1]=>[1,1,2,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,2,4,3,5,6,8,7]
[3,5]=>[1,1,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,2,4,3,5,6,7,8]
[4,1,1,1,1]=>[1,1,1,5]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,3,8,4,5,6,7]
[4,1,1,2]=>[1,1,1,4,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,2,3,7,4,5,6,8]
[4,1,2,1]=>[1,1,1,3,2]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,2,3,6,4,5,8,7]
[4,1,3]=>[1,1,1,3,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,2,3,6,4,5,7,8]
[4,2,1,1]=>[1,1,1,2,3]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,2,3,5,4,8,6,7]
[4,2,2]=>[1,1,1,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,3,5,4,7,6,8]
[4,3,1]=>[1,1,1,2,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,2,3,5,4,6,8,7]
[4,4]=>[1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,3,5,4,6,7,8]
[5,1,1,1]=>[1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,4,8,5,6,7]
[5,1,2]=>[1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,3,4,7,5,6,8]
[5,2,1]=>[1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,3,4,6,5,8,7]
[5,3]=>[1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,3,4,6,5,7,8]
[6,1,1]=>[1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,5,8,6,7]
[6,2]=>[1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,4,5,7,6,8]
[7,1]=>[1,1,1,1,1,1,2]=>[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]=>[1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7,8]
[1,1,1,1,1,1,1,1,1]=>[9]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[9,1,2,3,4,5,6,7,8]
[1,1,1,1,1,1,1,2]=>[8,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[8,1,2,3,4,5,6,7,9]
[1,1,1,1,1,1,2,1]=>[7,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]=>[7,1,2,3,4,5,6,9,8]
[1,1,1,1,1,1,3]=>[7,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>[7,1,2,3,4,5,6,8,9]
[1,1,1,1,1,2,1,1]=>[6,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]=>[6,1,2,3,4,5,9,7,8]
[1,1,1,1,1,2,2]=>[6,2,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]=>[6,1,2,3,4,5,8,7,9]
[1,1,1,1,1,3,1]=>[6,1,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]=>[6,1,2,3,4,5,7,9,8]
[1,1,1,1,1,4]=>[6,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]=>[6,1,2,3,4,5,7,8,9]
[1,1,1,1,2,1,1,1]=>[5,4]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[5,1,2,3,4,9,6,7,8]
[1,1,1,1,2,1,2]=>[5,3,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]=>[5,1,2,3,4,8,6,7,9]
[1,1,1,1,2,3]=>[5,2,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]=>[5,1,2,3,4,7,6,8,9]
[1,1,1,1,3,1,1]=>[5,1,3]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]=>[5,1,2,3,4,6,9,7,8]
[1,1,1,1,3,2]=>[5,1,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0]=>[5,1,2,3,4,6,8,7,9]
[1,1,1,1,4,1]=>[5,1,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]=>[5,1,2,3,4,6,7,9,8]
[1,1,1,1,5]=>[5,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]=>[5,1,2,3,4,6,7,8,9]
[1,1,1,2,1,1,1,1]=>[4,5]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[4,1,2,3,9,5,6,7,8]
[1,1,1,2,1,1,2]=>[4,4,1]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[4,1,2,3,8,5,6,7,9]
[1,1,1,2,1,2,1]=>[4,3,2]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]=>[4,1,2,3,7,5,6,9,8]
[1,1,1,2,1,3]=>[4,3,1,1]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]=>[4,1,2,3,7,5,6,8,9]
[1,1,1,2,2,1,1]=>[4,2,3]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0]=>[4,1,2,3,6,5,9,7,8]
[1,1,1,2,2,2]=>[4,2,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[4,1,2,3,6,5,8,7,9]
[1,1,1,2,4]=>[4,2,1,1,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]=>[4,1,2,3,6,5,7,8,9]
[1,1,1,3,1,1,1]=>[4,1,4]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]=>[4,1,2,3,5,9,6,7,8]
[1,1,1,3,1,2]=>[4,1,3,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0,1,0]=>[4,1,2,3,5,8,6,7,9]
[1,1,1,4,1,1]=>[4,1,1,3]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,0,0,0]=>[4,1,2,3,5,6,9,7,8]
[1,1,1,5,1]=>[4,1,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[4,1,2,3,5,6,7,9,8]
[1,1,1,6]=>[4,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[4,1,2,3,5,6,7,8,9]
[1,1,2,1,1,1,1,1]=>[3,6]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[3,1,2,9,4,5,6,7,8]
[1,1,2,1,1,1,2]=>[3,5,1]=>[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[3,1,2,8,4,5,6,7,9]
[1,1,2,1,1,2,1]=>[3,4,2]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[3,1,2,7,4,5,6,9,8]
[1,1,2,1,2,1,1]=>[3,3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[3,1,2,6,4,5,9,7,8]
[1,1,2,1,4]=>[3,3,1,1,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[3,1,2,6,4,5,7,8,9]
[1,1,2,2,2,1]=>[3,2,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[3,1,2,5,4,7,6,9,8]
[1,1,2,5]=>[3,2,1,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[3,1,2,5,4,6,7,8,9]
[1,1,3,1,1,1,1]=>[3,1,5]=>[1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[3,1,2,4,9,5,6,7,8]
[1,1,3,1,1,2]=>[3,1,4,1]=>[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[3,1,2,4,8,5,6,7,9]
[1,1,3,2,2]=>[3,1,2,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[3,1,2,4,6,5,8,7,9]
[1,1,3,3,1]=>[3,1,2,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[3,1,2,4,6,5,7,9,8]
[1,1,5,1,1]=>[3,1,1,1,3]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[3,1,2,4,5,6,9,7,8]
[1,1,5,2]=>[3,1,1,1,2,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[3,1,2,4,5,6,8,7,9]
[1,1,6,1]=>[3,1,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[3,1,2,4,5,6,7,9,8]
[1,1,7]=>[3,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[3,1,2,4,5,6,7,8,9]
[1,2,1,1,1,1,1,1]=>[2,7]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,1,9,3,4,5,6,7,8]
[1,2,1,1,1,1,2]=>[2,6,1]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[2,1,8,3,4,5,6,7,9]
[1,2,1,1,1,2,1]=>[2,5,2]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[2,1,7,3,4,5,6,9,8]
[1,2,1,1,1,3]=>[2,5,1,1]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[2,1,7,3,4,5,6,8,9]
[1,2,1,1,2,1,1]=>[2,4,3]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[2,1,6,3,4,5,9,7,8]
[1,2,1,1,4]=>[2,4,1,1,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[2,1,6,3,4,5,7,8,9]
[1,2,1,2,1,1,1]=>[2,3,4]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[2,1,5,3,4,9,6,7,8]
[1,2,1,2,1,2]=>[2,3,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[2,1,5,3,4,8,6,7,9]
[1,2,1,5]=>[2,3,1,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[2,1,5,3,4,6,7,8,9]
[1,2,2,1,1,1,1]=>[2,2,5]=>[1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,4,3,9,5,6,7,8]
[1,2,2,1,2,1]=>[2,2,3,2]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[2,1,4,3,7,5,6,9,8]
[1,2,2,1,3]=>[2,2,3,1,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[2,1,4,3,7,5,6,8,9]
[1,2,2,2,1,1]=>[2,2,2,3]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,4,3,6,5,9,7,8]
[1,2,2,2,2]=>[2,2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5,8,7,9]
[1,2,2,3,1]=>[2,2,2,1,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[2,1,4,3,6,5,7,9,8]
[1,2,2,4]=>[2,2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,4,3,6,5,7,8,9]
[1,2,3,1,1,1]=>[2,2,1,4]=>[1,1,0,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[2,1,4,3,5,9,6,7,8]
[1,2,5,1]=>[2,2,1,1,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,4,3,5,6,7,9,8]
[1,2,6]=>[2,2,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,4,3,5,6,7,8,9]
[1,3,1,1,1,1,1]=>[2,1,6]=>[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,1,3,9,4,5,6,7,8]
[1,3,1,1,1,2]=>[2,1,5,1]=>[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[2,1,3,8,4,5,6,7,9]
[1,3,1,2,1,1]=>[2,1,3,3]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[2,1,3,6,4,5,9,7,8]
[1,3,2,2,1]=>[2,1,2,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,3,5,4,7,6,9,8]
[1,3,5]=>[2,1,2,1,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,3,5,4,6,7,8,9]
[1,4,1,1,1,1]=>[2,1,1,5]=>[1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,3,4,9,5,6,7,8]
[1,4,1,1,2]=>[2,1,1,4,1]=>[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[2,1,3,4,8,5,6,7,9]
[1,4,2,1,1]=>[2,1,1,2,3]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,3,4,6,5,9,7,8]
[1,4,4]=>[2,1,1,2,1,1,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,3,4,6,5,7,8,9]
[1,5,1,1,1]=>[2,1,1,1,4]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[2,1,3,4,5,9,6,7,8]
[1,5,1,2]=>[2,1,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[2,1,3,4,5,8,6,7,9]
[1,5,2,1]=>[2,1,1,1,2,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[2,1,3,4,5,7,6,9,8]
[1,5,3]=>[2,1,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[2,1,3,4,5,7,6,8,9]
[1,6,1,1]=>[2,1,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[2,1,3,4,5,6,9,7,8]
[1,6,2]=>[2,1,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[2,1,3,4,5,6,8,7,9]
[1,7,1]=>[2,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,6,7,9,8]
[1,8]=>[2,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6,7,8,9]
[2,1,1,1,1,1,1,1]=>[1,8]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,9,2,3,4,5,6,7,8]
[2,1,1,1,1,1,2]=>[1,7,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[1,8,2,3,4,5,6,7,9]
[2,1,1,1,1,2,1]=>[1,6,2]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[1,7,2,3,4,5,6,9,8]
[2,1,1,1,1,3]=>[1,6,1,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[1,7,2,3,4,5,6,8,9]
[2,1,1,1,2,1,1]=>[1,5,3]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]=>[1,6,2,3,4,5,9,7,8]
[2,1,1,1,4]=>[1,5,1,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[1,6,2,3,4,5,7,8,9]
[2,1,1,2,1,1,1]=>[1,4,4]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[1,5,2,3,4,9,6,7,8]
[2,1,1,2,3]=>[1,4,2,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]=>[1,5,2,3,4,7,6,8,9]
[2,1,1,3,1,1]=>[1,4,1,3]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]=>[1,5,2,3,4,6,9,7,8]
[2,1,1,5]=>[1,4,1,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[1,5,2,3,4,6,7,8,9]
[2,1,2,1,1,1,1]=>[1,3,5]=>[1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,4,2,3,9,5,6,7,8]
[2,1,2,1,2,1]=>[1,3,3,2]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]=>[1,4,2,3,7,5,6,9,8]
[2,1,2,1,3]=>[1,3,3,1,1]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]=>[1,4,2,3,7,5,6,8,9]
[2,1,2,2,2]=>[1,3,2,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,4,2,3,6,5,8,7,9]
[2,1,6]=>[1,3,1,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,4,2,3,5,6,7,8,9]
[2,2,1,1,1,1,1]=>[1,2,6]=>[1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,3,2,9,4,5,6,7,8]
[2,2,1,1,1,2]=>[1,2,5,1]=>[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,3,2,8,4,5,6,7,9]
[2,2,1,2,1,1]=>[1,2,3,3]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,3,2,6,4,5,9,7,8]
[2,2,1,4]=>[1,2,3,1,1,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,3,2,6,4,5,7,8,9]
[2,2,2,1,1,1]=>[1,2,2,4]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,3,2,5,4,9,6,7,8]
[2,2,2,1,2]=>[1,2,2,3,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[1,3,2,5,4,8,6,7,9]
[2,2,2,2,1]=>[1,2,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4,7,6,9,8]
[2,2,2,3]=>[1,2,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,3,2,5,4,7,6,8,9]
[2,2,4,1]=>[1,2,2,1,1,2]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[1,3,2,5,4,6,7,9,8]
[2,2,5]=>[1,2,2,1,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,3,2,5,4,6,7,8,9]
[2,3,1,1,1,1]=>[1,2,1,5]=>[1,0,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,3,2,4,9,5,6,7,8]
[2,3,4]=>[1,2,1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,2,4,6,5,7,8,9]
[2,4,3]=>[1,2,1,1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,3,2,4,5,7,6,8,9]
[2,5,1,1]=>[1,2,1,1,1,3]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,3,2,4,5,6,9,7,8]
[2,5,2]=>[1,2,1,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,5,6,8,7,9]
[2,6,1]=>[1,2,1,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,3,2,4,5,6,7,9,8]
[2,7]=>[1,2,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,3,2,4,5,6,7,8,9]
[3,1,1,1,1,1,1]=>[1,1,7]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,2,9,3,4,5,6,7,8]
[3,1,1,1,1,2]=>[1,1,6,1]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[1,2,8,3,4,5,6,7,9]
[3,1,1,1,2,1]=>[1,1,5,2]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[1,2,7,3,4,5,6,9,8]
[3,1,1,1,3]=>[1,1,5,1,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[1,2,7,3,4,5,6,8,9]
[3,1,1,2,1,1]=>[1,1,4,3]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[1,2,6,3,4,5,9,7,8]
[3,1,1,4]=>[1,1,4,1,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[1,2,6,3,4,5,7,8,9]
[3,1,2,2,1]=>[1,1,3,2,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[1,2,5,3,4,7,6,9,8]
[3,1,5]=>[1,1,3,1,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[1,2,5,3,4,6,7,8,9]
[3,2,1,1,1,1]=>[1,1,2,5]=>[1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,4,3,9,5,6,7,8]
[3,2,1,1,2]=>[1,1,2,4,1]=>[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[1,2,4,3,8,5,6,7,9]
[3,2,1,2,1]=>[1,1,2,3,2]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[1,2,4,3,7,5,6,9,8]
[3,2,2,1,1]=>[1,1,2,2,3]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[1,2,4,3,6,5,9,7,8]
[3,2,2,2]=>[1,1,2,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,4,3,6,5,8,7,9]
[3,2,4]=>[1,1,2,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,4,3,6,5,7,8,9]
[3,3,2,1]=>[1,1,2,1,2,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,5,7,6,9,8]
[3,4,1,1]=>[1,1,2,1,1,3]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,4,3,5,6,9,7,8]
[3,4,2]=>[1,1,2,1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,4,3,5,6,8,7,9]
[3,5,1]=>[1,1,2,1,1,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,4,3,5,6,7,9,8]
[3,6]=>[1,1,2,1,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,4,3,5,6,7,8,9]
[4,1,1,1,1,1]=>[1,1,1,6]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,3,9,4,5,6,7,8]
[4,1,1,1,2]=>[1,1,1,5,1]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,2,3,8,4,5,6,7,9]
[4,1,2,1,1]=>[1,1,1,3,3]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,2,3,6,4,5,9,7,8]
[4,1,2,2]=>[1,1,1,3,2,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[1,2,3,6,4,5,8,7,9]
[4,1,4]=>[1,1,1,3,1,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,2,3,6,4,5,7,8,9]
[4,2,1,1,1]=>[1,1,1,2,4]=>[1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[1,2,3,5,4,9,6,7,8]
[4,2,2,1]=>[1,1,1,2,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,2,3,5,4,7,6,9,8]
[4,2,3]=>[1,1,1,2,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[1,2,3,5,4,7,6,8,9]
[4,3,2]=>[1,1,1,2,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[1,2,3,5,4,6,8,7,9]
[4,5]=>[1,1,1,2,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,2,3,5,4,6,7,8,9]
[5,1,1,1,1]=>[1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,3,4,9,5,6,7,8]
[5,1,1,2]=>[1,1,1,1,4,1]=>[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,2,3,4,8,5,6,7,9]
[5,1,2,1]=>[1,1,1,1,3,2]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[1,2,3,4,7,5,6,9,8]
[5,1,3]=>[1,1,1,1,3,1,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[1,2,3,4,7,5,6,8,9]
[5,2,1,1]=>[1,1,1,1,2,3]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[1,2,3,4,6,5,9,7,8]
[5,2,2]=>[1,1,1,1,2,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,3,4,6,5,8,7,9]
[5,3,1]=>[1,1,1,1,2,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[1,2,3,4,6,5,7,9,8]
[5,4]=>[1,1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,3,4,6,5,7,8,9]
[6,1,1,1]=>[1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,4,5,9,6,7,8]
[6,1,2]=>[1,1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,3,4,5,8,6,7,9]
[6,2,1]=>[1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,3,4,5,7,6,9,8]
[6,3]=>[1,1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,3,4,5,7,6,8,9]
[7,1,1]=>[1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,5,6,9,7,8]
[7,2]=>[1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,4,5,6,8,7,9]
[8,1]=>[1,1,1,1,1,1,1,2]=>[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]=>[1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7,8,9]
[1,1,1,1,1,1,1,1,1,1]=>[10]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[10,1,2,3,4,5,6,7,8,9]
[1,1,1,1,1,1,1,1,2]=>[9,1]=>[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]=>[9,1,2,3,4,5,6,7,8,10]
[1,1,1,1,1,1,1,2,1]=>[8,2]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]=>[8,1,2,3,4,5,6,7,10,9]
[1,1,1,1,1,1,1,3]=>[8,1,1]=>[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]=>[8,1,2,3,4,5,6,7,9,10]
[1,1,1,1,1,1,2,1,1]=>[7,3]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]=>[7,1,2,3,4,5,6,10,8,9]
[1,1,1,1,1,1,2,2]=>[7,2,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]=>[7,1,2,3,4,5,6,9,8,10]
[1,1,1,1,1,1,3,1]=>[7,1,2]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,1,0,0]=>[7,1,2,3,4,5,6,8,10,9]
[1,1,1,1,1,1,4]=>[7,1,1,1]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]=>[7,1,2,3,4,5,6,8,9,10]
[1,1,1,1,1,2,1,1,1]=>[6,4]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[6,1,2,3,4,5,10,7,8,9]
[1,1,1,1,1,2,1,2]=>[6,3,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]=>[6,1,2,3,4,5,9,7,8,10]
[1,1,1,1,1,3,1,1]=>[6,1,3]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,1,0,0,0]=>[6,1,2,3,4,5,7,10,8,9]
[1,1,1,1,1,3,2]=>[6,1,2,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0,1,0]=>[6,1,2,3,4,5,7,9,8,10]
[1,1,1,1,1,4,1]=>[6,1,1,2]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,1,0,0]=>[6,1,2,3,4,5,7,8,10,9]
[1,1,1,1,1,5]=>[6,1,1,1,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]=>[6,1,2,3,4,5,7,8,9,10]
[1,1,1,1,2,1,1,1,1]=>[5,5]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[5,1,2,3,4,10,6,7,8,9]
[1,1,1,1,2,1,1,2]=>[5,4,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]=>[5,1,2,3,4,9,6,7,8,10]
[1,1,1,1,2,2,2]=>[5,2,2,1]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]=>[5,1,2,3,4,7,6,9,8,10]
[1,1,1,1,3,1,1,1]=>[5,1,4]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,1,0,0,0,0]=>[5,1,2,3,4,6,10,7,8,9]
[1,1,1,1,3,1,2]=>[5,1,3,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0,1,0]=>[5,1,2,3,4,6,9,7,8,10]
[1,1,1,1,5,1]=>[5,1,1,1,2]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,1,0,0]=>[5,1,2,3,4,6,7,8,10,9]
[1,1,1,1,6]=>[5,1,1,1,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[5,1,2,3,4,6,7,8,9,10]
[1,1,1,2,1,1,1,1,1]=>[4,6]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[4,1,2,3,10,5,6,7,8,9]
[1,1,1,2,1,1,1,2]=>[4,5,1]=>[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[4,1,2,3,9,5,6,7,8,10]
[1,1,1,2,1,1,2,1]=>[4,4,2]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[4,1,2,3,8,5,6,7,10,9]
[1,1,1,2,1,1,3]=>[4,4,1,1]=>[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[4,1,2,3,8,5,6,7,9,10]
[1,1,1,2,1,2,1,1]=>[4,3,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[4,1,2,3,7,5,6,10,8,9]
[1,1,1,2,2,1,1,1]=>[4,2,4]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[4,1,2,3,6,5,10,7,8,9]
[1,1,1,2,2,2,1]=>[4,2,2,2]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[4,1,2,3,6,5,8,7,10,9]
[1,1,1,3,1,1,2]=>[4,1,4,1]=>[1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[4,1,2,3,5,9,6,7,8,10]
[1,1,1,4,1,1,1]=>[4,1,1,4]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[4,1,2,3,5,6,10,7,8,9]
[1,1,1,6,1]=>[4,1,1,1,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[4,1,2,3,5,6,7,8,10,9]
[1,1,1,7]=>[4,1,1,1,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[4,1,2,3,5,6,7,8,9,10]
[1,1,2,1,1,1,1,1,1]=>[3,7]=>[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[3,1,2,10,4,5,6,7,8,9]
[1,1,2,1,1,1,1,2]=>[3,6,1]=>[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[3,1,2,9,4,5,6,7,8,10]
[1,1,2,1,1,2,1,1]=>[3,4,3]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[3,1,2,7,4,5,6,10,8,9]
[1,1,2,1,2,1,1,1]=>[3,3,4]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[3,1,2,6,4,5,10,7,8,9]
[1,1,2,1,2,2,1]=>[3,3,2,2]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[3,1,2,6,4,5,8,7,10,9]
[1,1,2,2,2,1,1]=>[3,2,2,3]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[3,1,2,5,4,7,6,10,8,9]
[1,1,2,2,2,2]=>[3,2,2,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[3,1,2,5,4,7,6,9,8,10]
[1,1,2,6]=>[3,2,1,1,1,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[3,1,2,5,4,6,7,8,9,10]
[1,1,3,1,1,1,2]=>[3,1,5,1]=>[1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[3,1,2,4,9,5,6,7,8,10]
[1,1,6,2]=>[3,1,1,1,1,2,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[3,1,2,4,5,6,7,9,8,10]
[1,1,7,1]=>[3,1,1,1,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[3,1,2,4,5,6,7,8,10,9]
[1,1,8]=>[3,1,1,1,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[3,1,2,4,5,6,7,8,9,10]
[1,2,1,1,1,1,1,1,1]=>[2,8]=>[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[2,1,10,3,4,5,6,7,8,9]
[1,2,1,1,1,1,1,2]=>[2,7,1]=>[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[2,1,9,3,4,5,6,7,8,10]
[1,2,1,1,1,1,2,1]=>[2,6,2]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[2,1,8,3,4,5,6,7,10,9]
[1,2,1,1,1,1,3]=>[2,6,1,1]=>[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]=>[2,1,8,3,4,5,6,7,9,10]
[1,2,1,1,2,1,1,1]=>[2,4,4]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[2,1,6,3,4,5,10,7,8,9]
[1,2,1,1,2,1,2]=>[2,4,3,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]=>[2,1,6,3,4,5,9,7,8,10]
[1,2,1,2,1,2,1]=>[2,3,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]=>[2,1,5,3,4,8,6,7,10,9]
[1,2,1,6]=>[2,3,1,1,1,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,5,3,4,6,7,8,9,10]
[1,2,2,1,2,1,1]=>[2,2,3,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[2,1,4,3,7,5,6,10,8,9]
[1,2,2,1,4]=>[2,2,3,1,1,1]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[2,1,4,3,7,5,6,8,9,10]
[1,2,2,2,1,1,1]=>[2,2,2,4]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,4,3,6,5,10,7,8,9]
[1,2,2,2,1,2]=>[2,2,2,3,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[2,1,4,3,6,5,9,7,8,10]
[1,2,2,2,2,1]=>[2,2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5,8,7,10,9]
[1,2,2,2,3]=>[2,2,2,2,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[2,1,4,3,6,5,8,7,9,10]
[1,2,2,3,2]=>[2,2,2,1,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[2,1,4,3,6,5,7,9,8,10]
[1,2,2,5]=>[2,2,2,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,4,3,6,5,7,8,9,10]
[1,2,4,2,1]=>[2,2,1,1,2,2]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,5,6,8,7,10,9]
[1,2,7]=>[2,2,1,1,1,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,4,3,5,6,7,8,9,10]
[1,3,1,1,1,1,1,1]=>[2,1,7]=>[1,1,0,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[2,1,3,10,4,5,6,7,8,9]
[1,3,1,1,1,1,2]=>[2,1,6,1]=>[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[2,1,3,9,4,5,6,7,8,10]
[1,3,1,1,3,1]=>[2,1,4,1,2]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[2,1,3,7,4,5,6,8,10,9]
[1,3,6]=>[2,1,2,1,1,1,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,3,5,4,6,7,8,9,10]
[1,5,1,1,1,1]=>[2,1,1,1,5]=>[1,1,0,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,3,4,5,10,6,7,8,9]
[1,6,1,1,1]=>[2,1,1,1,1,4]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[2,1,3,4,5,6,10,7,8,9]
[1,6,1,2]=>[2,1,1,1,1,3,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[2,1,3,4,5,6,9,7,8,10]
[1,6,3]=>[2,1,1,1,1,2,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[2,1,3,4,5,6,8,7,9,10]
[1,7,1,1]=>[2,1,1,1,1,1,3]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[2,1,3,4,5,6,7,10,8,9]
[1,7,2]=>[2,1,1,1,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[2,1,3,4,5,6,7,9,8,10]
[1,8,1]=>[2,1,1,1,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,1,3,4,5,6,7,8,10,9]
[1,9]=>[2,1,1,1,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,1,3,4,5,6,7,8,9,10]
[2,1,1,1,1,1,1,1,1]=>[1,9]=>[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[1,10,2,3,4,5,6,7,8,9]
[2,1,1,1,1,1,1,2]=>[1,8,1]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]=>[1,9,2,3,4,5,6,7,8,10]
[2,1,1,1,1,1,2,1]=>[1,7,2]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]=>[1,8,2,3,4,5,6,7,10,9]
[2,1,1,1,1,1,3]=>[1,7,1,1]=>[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]=>[1,8,2,3,4,5,6,7,9,10]
[2,1,1,1,1,4]=>[1,6,1,1,1]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]=>[1,7,2,3,4,5,6,8,9,10]
[2,1,1,1,2,1,1,1]=>[1,5,4]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]=>[1,6,2,3,4,5,10,7,8,9]
[2,1,1,1,5]=>[1,5,1,1,1,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]=>[1,6,2,3,4,5,7,8,9,10]
[2,1,1,2,1,1,1,1]=>[1,4,5]=>[1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,5,2,3,4,10,6,7,8,9]
[2,1,1,6]=>[1,4,1,1,1,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,5,2,3,4,6,7,8,9,10]
[2,1,2,1,1,1,1,1]=>[1,3,6]=>[1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,4,2,3,10,5,6,7,8,9]
[2,1,2,1,2,1,1]=>[1,3,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[1,4,2,3,7,5,6,10,8,9]
[2,1,2,2,2,1]=>[1,3,2,2,2]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,4,2,3,6,5,8,7,10,9]
[2,1,7]=>[1,3,1,1,1,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,4,2,3,5,6,7,8,9,10]
[2,2,1,1,1,1,1,1]=>[1,2,7]=>[1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,3,2,10,4,5,6,7,8,9]
[2,2,1,1,4]=>[1,2,4,1,1,1]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[1,3,2,7,4,5,6,8,9,10]
[2,2,1,2,1,1,1]=>[1,2,3,4]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[1,3,2,6,4,5,10,7,8,9]
[2,2,1,2,2,1]=>[1,2,3,2,2]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,6,4,5,8,7,10,9]
[2,2,2,2,1,1]=>[1,2,2,2,3]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[1,3,2,5,4,7,6,10,8,9]
[2,2,2,2,2]=>[1,2,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,7,6,9,8,10]
[2,2,2,4]=>[1,2,2,2,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,2,5,4,7,6,8,9,10]
[2,2,6]=>[1,2,2,1,1,1,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,3,2,5,4,6,7,8,9,10]
[2,3,1,3,1]=>[1,2,1,3,1,2]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[1,3,2,4,7,5,6,8,10,9]
[2,6,1,1]=>[1,2,1,1,1,1,3]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,3,2,4,5,6,7,10,8,9]
[2,6,2]=>[1,2,1,1,1,1,2,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,3,2,4,5,6,7,9,8,10]
[2,7,1]=>[1,2,1,1,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,3,2,4,5,6,7,8,10,9]
[2,8]=>[1,2,1,1,1,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,3,2,4,5,6,7,8,9,10]
[3,1,1,1,1,1,1,1]=>[1,1,8]=>[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[1,2,10,3,4,5,6,7,8,9]
[3,1,1,1,1,1,2]=>[1,1,7,1]=>[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]=>[1,2,9,3,4,5,6,7,8,10]
[3,1,1,1,1,2,1]=>[1,1,6,2]=>[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]=>[1,2,8,3,4,5,6,7,10,9]
[3,1,1,1,4]=>[1,1,5,1,1,1]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]=>[1,2,7,3,4,5,6,8,9,10]
[3,1,1,2,1,1,1]=>[1,1,4,4]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]=>[1,2,6,3,4,5,10,7,8,9]
[3,1,1,5]=>[1,1,4,1,1,1,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]=>[1,2,6,3,4,5,7,8,9,10]
[3,1,6]=>[1,1,3,1,1,1,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,5,3,4,6,7,8,9,10]
[3,2,2,2,1]=>[1,1,2,2,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,6,5,8,7,10,9]
[3,2,5]=>[1,1,2,2,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,2,4,3,6,5,7,8,9,10]
[3,7]=>[1,1,2,1,1,1,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,4,3,5,6,7,8,9,10]
[4,1,1,1,1,1,1]=>[1,1,1,7]=>[1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,2,3,10,4,5,6,7,8,9]
[4,1,1,1,1,2]=>[1,1,1,6,1]=>[1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[1,2,3,9,4,5,6,7,8,10]
[4,1,1,2,2]=>[1,1,1,4,2,1]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[1,2,3,7,4,5,6,9,8,10]
[4,1,5]=>[1,1,1,3,1,1,1,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[1,2,3,6,4,5,7,8,9,10]
[4,2,2,2]=>[1,1,1,2,2,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,3,5,4,7,6,9,8,10]
[4,2,4]=>[1,1,1,2,2,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,3,5,4,7,6,8,9,10]
[4,5,1]=>[1,1,1,2,1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[1,2,3,5,4,6,7,8,10,9]
[4,6]=>[1,1,1,2,1,1,1,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,5,4,6,7,8,9,10]
[5,1,1,1,1,1]=>[1,1,1,1,6]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,3,4,10,5,6,7,8,9]
[5,1,1,1,2]=>[1,1,1,1,5,1]=>[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[1,2,3,4,9,5,6,7,8,10]
[5,1,4]=>[1,1,1,1,3,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[1,2,3,4,7,5,6,8,9,10]
[5,2,2,1]=>[1,1,1,1,2,2,2]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[1,2,3,4,6,5,8,7,10,9]
[5,5]=>[1,1,1,1,2,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,6,5,7,8,9,10]
[6,1,1,1,1]=>[1,1,1,1,1,5]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[1,2,3,4,5,10,6,7,8,9]
[6,1,1,2]=>[1,1,1,1,1,4,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[1,2,3,4,5,9,6,7,8,10]
[6,2,2]=>[1,1,1,1,1,2,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,3,4,5,7,6,9,8,10]
[6,4]=>[1,1,1,1,1,2,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,3,4,5,7,6,8,9,10]
[7,1,1,1]=>[1,1,1,1,1,1,4]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[1,2,3,4,5,6,10,7,8,9]
[7,1,2]=>[1,1,1,1,1,1,3,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[1,2,3,4,5,6,9,7,8,10]
[7,2,1]=>[1,1,1,1,1,1,2,2]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[1,2,3,4,5,6,8,7,10,9]
[7,3]=>[1,1,1,1,1,1,2,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[1,2,3,4,5,6,8,7,9,10]
[8,1,1]=>[1,1,1,1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[1,2,3,4,5,6,7,10,8,9]
[8,2]=>[1,1,1,1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[1,2,3,4,5,6,7,9,8,10]
[9,1]=>[1,1,1,1,1,1,1,1,2]=>[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]=>[1,1,1,1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[1,2,3,4,5,6,7,8,9,10]
[10,1]=>[1,1,1,1,1,1,1,1,1,2]=>[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]
[2,2,2,2,2,2]=>[1,2,2,2,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,7,6,9,8,11,10,12]
[1,2,2,2,2,2,1]=>[2,2,2,2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5,8,7,10,9,12,11]
[1,1,1,1,1,1,1,1,1,2]=>[10,1]=>[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]=>[10,1,2,3,4,5,6,7,8,9,11]
[2,1,1,1,1,1,1,1,1,1]=>[1,10]=>[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[1,11,2,3,4,5,6,7,8,9,10]
[2,8,1]=>[1,2,1,1,1,1,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[1,3,2,4,5,6,7,8,9,11,10]
[2,1,1,1,1,1,1,2,1]=>[1,8,2]=>[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]=>[1,9,2,3,4,5,6,7,8,11,10]
Map
complement
Description
The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
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