Identifier
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Images
=>
Cc0005;cc-rep-2
[1]=>[1]=>[1,0]=>[1] [1,1]=>[2]=>[1,1,0,0]=>[1,2] [2]=>[1,1]=>[1,0,1,0]=>[2,1] [1,1,1]=>[3]=>[1,1,1,0,0,0]=>[1,2,3] [1,2]=>[2,1]=>[1,1,0,0,1,0]=>[1,3,2] [2,1]=>[1,2]=>[1,0,1,1,0,0]=>[2,1,3] [3]=>[1,1,1]=>[1,0,1,0,1,0]=>[2,3,1] [1,1,1,1]=>[4]=>[1,1,1,1,0,0,0,0]=>[1,2,3,4] [1,1,2]=>[3,1]=>[1,1,1,0,0,0,1,0]=>[1,2,4,3] [1,2,1]=>[2,2]=>[1,1,0,0,1,1,0,0]=>[1,3,2,4] [1,3]=>[2,1,1]=>[1,1,0,0,1,0,1,0]=>[1,3,4,2] [2,1,1]=>[1,3]=>[1,0,1,1,1,0,0,0]=>[2,1,3,4] [2,2]=>[1,2,1]=>[1,0,1,1,0,0,1,0]=>[2,1,4,3] [3,1]=>[1,1,2]=>[1,0,1,0,1,1,0,0]=>[2,3,1,4] [4]=>[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[2,3,4,1] [1,1,1,1,1]=>[5]=>[1,1,1,1,1,0,0,0,0,0]=>[1,2,3,4,5] [1,1,1,2]=>[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[1,2,3,5,4] [1,1,2,1]=>[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[1,2,4,3,5] [1,1,3]=>[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[1,2,4,5,3] [1,2,1,1]=>[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[1,3,2,4,5] [1,2,2]=>[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4] [1,3,1]=>[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[1,3,4,2,5] [1,4]=>[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[1,3,4,5,2] [2,1,1,1]=>[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[2,1,3,4,5] [2,1,2]=>[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[2,1,3,5,4] [2,2,1]=>[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,5] [2,3]=>[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[2,1,4,5,3] [3,1,1]=>[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[2,3,1,4,5] [3,2]=>[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[2,3,1,5,4] [4,1]=>[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[2,3,4,1,5] [5]=>[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,1] [1,1,1,1,1,1]=>[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[1,2,3,4,5,6] [1,1,1,1,2]=>[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[1,2,3,4,6,5] [1,1,1,2,1]=>[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[1,2,3,5,4,6] [1,1,1,3]=>[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[1,2,3,5,6,4] [1,1,2,1,1]=>[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[1,2,4,3,5,6] [1,1,2,2]=>[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[1,2,4,3,6,5] [1,1,3,1]=>[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[1,2,4,5,3,6] [1,1,4]=>[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[1,2,4,5,6,3] [1,2,1,1,1]=>[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[1,3,2,4,5,6] [1,2,1,2]=>[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[1,3,2,4,6,5] [1,2,2,1]=>[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[1,3,2,5,4,6] [1,2,3]=>[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[1,3,2,5,6,4] [1,3,1,1]=>[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[1,3,4,2,5,6] [1,3,2]=>[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[1,3,4,2,6,5] [1,4,1]=>[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[1,3,4,5,2,6] [1,5]=>[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[1,3,4,5,6,2] [2,1,1,1,1]=>[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[2,1,3,4,5,6] [2,1,1,2]=>[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[2,1,3,4,6,5] [2,1,2,1]=>[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[2,1,3,5,4,6] [2,1,3]=>[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[2,1,3,5,6,4] [2,2,1,1]=>[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[2,1,4,3,5,6] [2,2,2]=>[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[2,1,4,3,6,5] [2,3,1]=>[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[2,1,4,5,3,6] [2,4]=>[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[2,1,4,5,6,3] [3,1,1,1]=>[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[2,3,1,4,5,6] [3,1,2]=>[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[2,3,1,4,6,5] [3,2,1]=>[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[2,3,1,5,4,6] [3,3]=>[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[2,3,1,5,6,4] [4,1,1]=>[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[2,3,4,1,5,6] [4,2]=>[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[2,3,4,1,6,5] [5,1]=>[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[2,3,4,5,1,6] [6]=>[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,6,1] [1,1,1,1,1,1,1]=>[7]=>[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[1,2,3,4,5,6,7] [1,1,1,1,1,2]=>[6,1]=>[1,1,1,1,1,1,0,0,0,0,0,0,1,0]=>[1,2,3,4,5,7,6] [1,1,1,1,2,1]=>[5,2]=>[1,1,1,1,1,0,0,0,0,0,1,1,0,0]=>[1,2,3,4,6,5,7] [1,1,1,1,3]=>[5,1,1]=>[1,1,1,1,1,0,0,0,0,0,1,0,1,0]=>[1,2,3,4,6,7,5] [1,1,1,2,1,1]=>[4,3]=>[1,1,1,1,0,0,0,0,1,1,1,0,0,0]=>[1,2,3,5,4,6,7] [1,1,1,2,2]=>[4,2,1]=>[1,1,1,1,0,0,0,0,1,1,0,0,1,0]=>[1,2,3,5,4,7,6] [1,1,1,3,1]=>[4,1,2]=>[1,1,1,1,0,0,0,0,1,0,1,1,0,0]=>[1,2,3,5,6,4,7] [1,1,1,4]=>[4,1,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0,1,0]=>[1,2,3,5,6,7,4] [1,1,2,1,1,1]=>[3,4]=>[1,1,1,0,0,0,1,1,1,1,0,0,0,0]=>[1,2,4,3,5,6,7] [1,1,2,1,2]=>[3,3,1]=>[1,1,1,0,0,0,1,1,1,0,0,0,1,0]=>[1,2,4,3,5,7,6] [1,1,2,2,1]=>[3,2,2]=>[1,1,1,0,0,0,1,1,0,0,1,1,0,0]=>[1,2,4,3,6,5,7] [1,1,2,3]=>[3,2,1,1]=>[1,1,1,0,0,0,1,1,0,0,1,0,1,0]=>[1,2,4,3,6,7,5] [1,1,3,1,1]=>[3,1,3]=>[1,1,1,0,0,0,1,0,1,1,1,0,0,0]=>[1,2,4,5,3,6,7] [1,1,3,2]=>[3,1,2,1]=>[1,1,1,0,0,0,1,0,1,1,0,0,1,0]=>[1,2,4,5,3,7,6] [1,1,4,1]=>[3,1,1,2]=>[1,1,1,0,0,0,1,0,1,0,1,1,0,0]=>[1,2,4,5,6,3,7] [1,1,5]=>[3,1,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0,1,0]=>[1,2,4,5,6,7,3] [1,2,1,1,1,1]=>[2,5]=>[1,1,0,0,1,1,1,1,1,0,0,0,0,0]=>[1,3,2,4,5,6,7] [1,2,1,1,2]=>[2,4,1]=>[1,1,0,0,1,1,1,1,0,0,0,0,1,0]=>[1,3,2,4,5,7,6] [1,2,1,2,1]=>[2,3,2]=>[1,1,0,0,1,1,1,0,0,0,1,1,0,0]=>[1,3,2,4,6,5,7] [1,2,1,3]=>[2,3,1,1]=>[1,1,0,0,1,1,1,0,0,0,1,0,1,0]=>[1,3,2,4,6,7,5] [1,2,2,1,1]=>[2,2,3]=>[1,1,0,0,1,1,0,0,1,1,1,0,0,0]=>[1,3,2,5,4,6,7] [1,2,2,2]=>[2,2,2,1]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[1,3,2,5,4,7,6] [1,2,3,1]=>[2,2,1,2]=>[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[1,3,2,5,6,4,7] [1,2,4]=>[2,2,1,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[1,3,2,5,6,7,4] [1,3,1,1,1]=>[2,1,4]=>[1,1,0,0,1,0,1,1,1,1,0,0,0,0]=>[1,3,4,2,5,6,7] [1,3,1,2]=>[2,1,3,1]=>[1,1,0,0,1,0,1,1,1,0,0,0,1,0]=>[1,3,4,2,5,7,6] [1,3,2,1]=>[2,1,2,2]=>[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[1,3,4,2,6,5,7] [1,3,3]=>[2,1,2,1,1]=>[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[1,3,4,2,6,7,5] [1,4,1,1]=>[2,1,1,3]=>[1,1,0,0,1,0,1,0,1,1,1,0,0,0]=>[1,3,4,5,2,6,7] [1,4,2]=>[2,1,1,2,1]=>[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[1,3,4,5,2,7,6] [1,5,1]=>[2,1,1,1,2]=>[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[1,3,4,5,6,2,7] [1,6]=>[2,1,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[1,3,4,5,6,7,2] [2,1,1,1,1,1]=>[1,6]=>[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[2,1,3,4,5,6,7] [2,1,1,1,2]=>[1,5,1]=>[1,0,1,1,1,1,1,0,0,0,0,0,1,0]=>[2,1,3,4,5,7,6] [2,1,1,2,1]=>[1,4,2]=>[1,0,1,1,1,1,0,0,0,0,1,1,0,0]=>[2,1,3,4,6,5,7] [2,1,1,3]=>[1,4,1,1]=>[1,0,1,1,1,1,0,0,0,0,1,0,1,0]=>[2,1,3,4,6,7,5] [2,1,2,1,1]=>[1,3,3]=>[1,0,1,1,1,0,0,0,1,1,1,0,0,0]=>[2,1,3,5,4,6,7] [2,1,2,2]=>[1,3,2,1]=>[1,0,1,1,1,0,0,0,1,1,0,0,1,0]=>[2,1,3,5,4,7,6] [2,1,3,1]=>[1,3,1,2]=>[1,0,1,1,1,0,0,0,1,0,1,1,0,0]=>[2,1,3,5,6,4,7] [2,1,4]=>[1,3,1,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0,1,0]=>[2,1,3,5,6,7,4] [2,2,1,1,1]=>[1,2,4]=>[1,0,1,1,0,0,1,1,1,1,0,0,0,0]=>[2,1,4,3,5,6,7] [2,2,1,2]=>[1,2,3,1]=>[1,0,1,1,0,0,1,1,1,0,0,0,1,0]=>[2,1,4,3,5,7,6] [2,2,2,1]=>[1,2,2,2]=>[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[2,1,4,3,6,5,7] [2,2,3]=>[1,2,2,1,1]=>[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[2,1,4,3,6,7,5] [2,3,1,1]=>[1,2,1,3]=>[1,0,1,1,0,0,1,0,1,1,1,0,0,0]=>[2,1,4,5,3,6,7] [2,3,2]=>[1,2,1,2,1]=>[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[2,1,4,5,3,7,6] [2,4,1]=>[1,2,1,1,2]=>[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[2,1,4,5,6,3,7] [2,5]=>[1,2,1,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,4,5,6,7,3] [3,1,1,1,1]=>[1,1,5]=>[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[2,3,1,4,5,6,7] [3,1,1,2]=>[1,1,4,1]=>[1,0,1,0,1,1,1,1,0,0,0,0,1,0]=>[2,3,1,4,5,7,6] [3,1,2,1]=>[1,1,3,2]=>[1,0,1,0,1,1,1,0,0,0,1,1,0,0]=>[2,3,1,4,6,5,7] [3,1,3]=>[1,1,3,1,1]=>[1,0,1,0,1,1,1,0,0,0,1,0,1,0]=>[2,3,1,4,6,7,5] [3,2,1,1]=>[1,1,2,3]=>[1,0,1,0,1,1,0,0,1,1,1,0,0,0]=>[2,3,1,5,4,6,7] [3,2,2]=>[1,1,2,2,1]=>[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[2,3,1,5,4,7,6] [3,3,1]=>[1,1,2,1,2]=>[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[2,3,1,5,6,4,7] [3,4]=>[1,1,2,1,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[2,3,1,5,6,7,4] [4,1,1,1]=>[1,1,1,4]=>[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[2,3,4,1,5,6,7] [4,1,2]=>[1,1,1,3,1]=>[1,0,1,0,1,0,1,1,1,0,0,0,1,0]=>[2,3,4,1,5,7,6] [4,2,1]=>[1,1,1,2,2]=>[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[2,3,4,1,6,5,7] [4,3]=>[1,1,1,2,1,1]=>[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[2,3,4,1,6,7,5] [5,1,1]=>[1,1,1,1,3]=>[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[2,3,4,5,1,6,7] [5,2]=>[1,1,1,1,2,1]=>[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[2,3,4,5,1,7,6] [6,1]=>[1,1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[2,3,4,5,6,1,7] [7]=>[1,1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[2,3,4,5,6,7,1] [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]=>[1,2,3,4,5,6,7,8] [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]=>[1,2,3,4,5,6,8,7] [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]=>[1,2,3,4,5,7,6,8] [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]=>[1,2,3,4,5,7,8,6] [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]=>[1,2,3,4,6,5,7,8] [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]=>[1,2,3,4,6,5,8,7] [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]=>[1,2,3,4,6,7,5,8] [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]=>[1,2,3,4,6,7,8,5] [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]=>[1,2,3,5,4,6,7,8] [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]=>[1,2,3,5,4,6,8,7] [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]=>[1,2,3,5,4,7,6,8] [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]=>[1,2,3,5,4,7,8,6] [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]=>[1,2,3,5,6,4,7,8] [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]=>[1,2,3,5,6,4,8,7] [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]=>[1,2,3,5,6,7,4,8] [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]=>[1,2,3,5,6,7,8,4] [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]=>[1,2,4,3,5,6,7,8] [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]=>[1,2,4,3,5,6,8,7] [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]=>[1,2,4,3,5,7,6,8] [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]=>[1,2,4,3,5,7,8,6] [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]=>[1,2,4,3,6,5,7,8] [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]=>[1,2,4,3,6,5,8,7] [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]=>[1,2,4,3,6,7,5,8] [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]=>[1,2,4,3,6,7,8,5] [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]=>[1,2,4,5,3,6,7,8] [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]=>[1,2,4,5,3,6,8,7] [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]=>[1,2,4,5,3,7,6,8] [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]=>[1,2,4,5,3,7,8,6] [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]=>[1,2,4,5,6,3,7,8] [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]=>[1,2,4,5,6,3,8,7] [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]=>[1,2,4,5,6,7,3,8] [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]=>[1,2,4,5,6,7,8,3] [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]=>[1,3,2,4,5,6,7,8] [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]=>[1,3,2,4,5,6,8,7] [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]=>[1,3,2,4,5,7,6,8] [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]=>[1,3,2,4,5,7,8,6] [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]=>[1,3,2,4,6,5,7,8] [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]=>[1,3,2,4,6,5,8,7] [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]=>[1,3,2,4,6,7,5,8] [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]=>[1,3,2,4,6,7,8,5] [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]=>[1,3,2,5,4,6,7,8] [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]=>[1,3,2,5,4,6,8,7] [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]=>[1,3,2,5,4,7,6,8] [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]=>[1,3,2,5,4,7,8,6] [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]=>[1,3,2,5,6,4,7,8] [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]=>[1,3,2,5,6,4,8,7] [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]=>[1,3,2,5,6,7,4,8] [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]=>[1,3,2,5,6,7,8,4] [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]=>[1,3,4,2,5,6,7,8] [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]=>[1,3,4,2,5,6,8,7] [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]=>[1,3,4,2,5,7,6,8] [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]=>[1,3,4,2,5,7,8,6] [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]=>[1,3,4,2,6,5,7,8] [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]=>[1,3,4,2,6,5,8,7] [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]=>[1,3,4,2,6,7,5,8] [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]=>[1,3,4,2,6,7,8,5] [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]=>[1,3,4,5,2,6,7,8] [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]=>[1,3,4,5,2,6,8,7] [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]=>[1,3,4,5,2,7,6,8] [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]=>[1,3,4,5,2,7,8,6] [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]=>[1,3,4,5,6,2,7,8] [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]=>[1,3,4,5,6,2,8,7] [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]=>[1,3,4,5,6,7,2,8] [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]=>[1,3,4,5,6,7,8,2] [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]=>[2,1,3,4,5,6,7,8] [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]=>[2,1,3,4,5,6,8,7] [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]=>[2,1,3,4,5,7,6,8] [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]=>[2,1,3,4,5,7,8,6] [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]=>[2,1,3,4,6,5,7,8] [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]=>[2,1,3,4,6,5,8,7] [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]=>[2,1,3,4,6,7,5,8] [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]=>[2,1,3,4,6,7,8,5] [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]=>[2,1,3,5,4,6,7,8] [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]=>[2,1,3,5,4,6,8,7] [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]=>[2,1,3,5,4,7,6,8] [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]=>[2,1,3,5,4,7,8,6] [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]=>[2,1,3,5,6,4,7,8] [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]=>[2,1,3,5,6,4,8,7] [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]=>[2,1,3,5,6,7,4,8] [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]=>[2,1,3,5,6,7,8,4] [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]=>[2,1,4,3,5,6,7,8] [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]=>[2,1,4,3,5,6,8,7] [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]=>[2,1,4,3,5,7,6,8] [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]=>[2,1,4,3,5,7,8,6] [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]=>[2,1,4,3,6,5,7,8] [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]=>[2,1,4,3,6,5,8,7] [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]=>[2,1,4,3,6,7,5,8] [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]=>[2,1,4,3,6,7,8,5] [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]=>[2,1,4,5,3,6,7,8] [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]=>[2,1,4,5,3,6,8,7] [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]=>[2,1,4,5,3,7,6,8] [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]=>[2,1,4,5,3,7,8,6] [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]=>[2,1,4,5,6,3,7,8] [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]=>[2,1,4,5,6,3,8,7] [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]=>[2,1,4,5,6,7,3,8] [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]=>[2,1,4,5,6,7,8,3] [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]=>[2,3,1,4,5,6,7,8] [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]=>[2,3,1,4,5,6,8,7] [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]=>[2,3,1,4,5,7,6,8] [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]=>[2,3,1,4,5,7,8,6] [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]=>[2,3,1,4,6,5,7,8] [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]=>[2,3,1,4,6,5,8,7] [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]=>[2,3,1,4,6,7,5,8] [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]=>[2,3,1,4,6,7,8,5] [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]=>[2,3,1,5,4,6,7,8] [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]=>[2,3,1,5,4,6,8,7] [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]=>[2,3,1,5,4,7,6,8] [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]=>[2,3,1,5,4,7,8,6] [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]=>[2,3,1,5,6,4,7,8] [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]=>[2,3,1,5,6,4,8,7] [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]=>[2,3,1,5,6,7,4,8] [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]=>[2,3,1,5,6,7,8,4] [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]=>[2,3,4,1,5,6,7,8] [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]=>[2,3,4,1,5,6,8,7] [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]=>[2,3,4,1,5,7,6,8] [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]=>[2,3,4,1,5,7,8,6] [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]=>[2,3,4,1,6,5,7,8] [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]=>[2,3,4,1,6,5,8,7] [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]=>[2,3,4,1,6,7,5,8] [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]=>[2,3,4,1,6,7,8,5] [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]=>[2,3,4,5,1,6,7,8] [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]=>[2,3,4,5,1,6,8,7] [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]=>[2,3,4,5,1,7,6,8] [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]=>[2,3,4,5,1,7,8,6] [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]=>[2,3,4,5,6,1,7,8] [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]=>[2,3,4,5,6,1,8,7] [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]=>[2,3,4,5,6,7,1,8] [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]=>[2,3,4,5,6,7,8,1] [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]=>[1,2,3,4,5,6,7,8,9] [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]=>[1,2,3,4,5,6,7,9,8] [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]=>[1,2,3,4,5,6,8,7,9] [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]=>[1,2,3,4,5,6,8,9,7] [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]=>[1,2,3,4,5,7,6,8,9] [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]=>[1,2,3,4,6,5,7,8,9] [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]=>[1,2,3,5,4,6,7,8,9] [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]=>[1,2,4,3,5,6,7,8,9] [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]=>[1,2,4,5,3,6,7,8,9] [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]=>[1,2,4,5,6,7,8,9,3] [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]=>[1,3,2,4,5,6,7,8,9] [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]=>[1,3,2,5,6,7,8,9,4] [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]=>[1,3,4,2,6,7,8,9,5] [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]=>[1,3,4,5,2,7,8,9,6] [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]=>[1,3,4,5,6,2,8,7,9] [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]=>[1,3,4,5,6,2,8,9,7] [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]=>[1,3,4,5,6,7,2,9,8] [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]=>[1,3,4,5,6,7,8,2,9] [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]=>[1,3,4,5,6,7,8,9,2] [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]=>[2,1,3,4,5,6,7,8,9] [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]=>[2,1,3,4,5,6,7,9,8] [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]=>[2,1,4,3,5,6,7,8,9] [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]=>[2,1,4,5,6,7,8,9,3] [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]=>[2,3,1,4,5,6,7,8,9] [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]=>[2,3,1,4,5,6,7,9,8] [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]=>[2,3,1,5,4,6,7,8,9] [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]=>[2,3,1,5,6,7,8,9,4] [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]=>[2,3,4,1,5,6,7,8,9] [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]=>[2,3,4,1,6,5,7,8,9] [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]=>[2,3,4,1,6,7,8,9,5] [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]=>[2,3,4,5,1,6,7,8,9] [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]=>[2,3,4,5,1,7,6,8,9] [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]=>[2,3,4,5,1,7,8,9,6] [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]=>[2,3,4,5,6,1,7,8,9] [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]=>[2,3,4,5,6,1,7,9,8] [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]=>[2,3,4,5,6,1,8,7,9] [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]=>[2,3,4,5,6,1,8,9,7] [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]=>[2,3,4,5,6,7,1,8,9] [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]=>[2,3,4,5,6,7,1,9,8] [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]=>[2,3,4,5,6,7,8,1,9] [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]=>[2,3,4,5,6,7,8,9,1] [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]=>[1,2,3,4,5,6,7,8,9,10] [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]=>[1,2,3,4,5,6,7,8,10,9] [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]=>[1,2,3,4,5,6,7,9,8,10] [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]=>[1,2,3,4,5,6,8,7,9,10] [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]=>[1,2,3,4,5,7,6,8,9,10] [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]=>[1,2,3,4,6,5,7,8,9,10] [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]=>[1,2,3,5,4,6,7,8,9,10] [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]=>[1,2,4,3,5,6,7,8,9,10] [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]=>[1,3,2,4,5,6,7,8,9,10] [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]=>[1,3,4,5,6,7,8,9,2,10] [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]=>[1,3,4,5,6,7,8,9,10,2] [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]=>[2,1,3,4,5,6,7,8,9,10] [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]=>[2,1,3,4,5,6,7,8,10,9] [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]=>[2,1,4,5,6,7,8,9,10,3] [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]=>[2,3,1,4,5,6,7,8,9,10] [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]=>[2,3,1,5,6,7,8,9,10,4] [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]=>[2,3,4,1,5,6,7,8,9,10] [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]=>[2,3,4,1,6,7,8,9,10,5] [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]=>[2,3,4,5,1,6,7,8,9,10] [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]=>[2,3,4,5,1,7,8,9,10,6] [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]=>[2,3,4,5,6,1,7,8,9,10] [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]=>[2,3,4,5,6,1,8,9,10,7] [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]=>[2,3,4,5,6,7,1,8,9,10] [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]=>[2,3,4,5,6,7,1,9,10,8] [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]=>[2,3,4,5,6,7,8,1,9,10] [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]=>[2,3,4,5,6,7,8,1,10,9] [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]=>[2,3,4,5,6,7,8,9,1,10] [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]=>[2,3,4,5,6,7,8,9,10,1]
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 $I$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.