Identifier
Mp00035: Dyck paths to alternating sign matrixAlternating sign matrices
Mp00002: Alternating sign matrices to left key permutationPermutations
Mp00066: Permutations inversePermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Images
=>
Cc0005;cc-rep-0Cc0017;cc-rep-1
[1,0]=>[[1]]=>[1]=>[1]=>[1] [1,0,1,0]=>[[1,0],[0,1]]=>[1,2]=>[1,2]=>[1,2] [1,1,0,0]=>[[0,1],[1,0]]=>[2,1]=>[2,1]=>[2,1] [1,0,1,0,1,0]=>[[1,0,0],[0,1,0],[0,0,1]]=>[1,2,3]=>[1,2,3]=>[1,2,3] [1,0,1,1,0,0]=>[[1,0,0],[0,0,1],[0,1,0]]=>[1,3,2]=>[1,3,2]=>[1,3,2] [1,1,0,0,1,0]=>[[0,1,0],[1,0,0],[0,0,1]]=>[2,1,3]=>[2,1,3]=>[2,1,3] [1,1,0,1,0,0]=>[[0,1,0],[1,-1,1],[0,1,0]]=>[1,3,2]=>[1,3,2]=>[1,3,2] [1,1,1,0,0,0]=>[[0,0,1],[1,0,0],[0,1,0]]=>[3,1,2]=>[2,3,1]=>[3,1,2] [1,0,1,0,1,0,1,0]=>[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4] [1,0,1,0,1,1,0,0]=>[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]=>[1,2,4,3]=>[1,2,4,3]=>[1,2,4,3] [1,0,1,1,0,0,1,0]=>[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]=>[1,3,2,4]=>[1,3,2,4]=>[1,3,2,4] [1,0,1,1,0,1,0,0]=>[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]=>[1,2,4,3]=>[1,2,4,3]=>[1,2,4,3] [1,0,1,1,1,0,0,0]=>[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]=>[1,4,2,3]=>[1,3,4,2]=>[1,4,2,3] [1,1,0,0,1,0,1,0]=>[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]=>[2,1,3,4]=>[2,1,3,4]=>[2,1,3,4] [1,1,0,0,1,1,0,0]=>[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]=>[2,1,4,3]=>[2,1,4,3]=>[2,1,4,3] [1,1,0,1,0,0,1,0]=>[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]=>[1,3,2,4]=>[1,3,2,4]=>[1,3,2,4] [1,1,0,1,0,1,0,0]=>[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]=>[1,2,4,3]=>[1,2,4,3]=>[1,2,4,3] [1,1,0,1,1,0,0,0]=>[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]=>[1,4,2,3]=>[1,3,4,2]=>[1,4,2,3] [1,1,1,0,0,0,1,0]=>[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]=>[3,1,2,4]=>[2,3,1,4]=>[3,1,2,4] [1,1,1,0,0,1,0,0]=>[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]=>[2,1,4,3]=>[2,1,4,3]=>[2,1,4,3] [1,1,1,0,1,0,0,0]=>[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]=>[1,4,2,3]=>[1,3,4,2]=>[1,4,2,3] [1,1,1,1,0,0,0,0]=>[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]=>[4,1,2,3]=>[2,3,4,1]=>[4,1,2,3] [1,0,1,0,1,0,1,0,1,0]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[1,2,3,4,5]=>[1,2,3,4,5]=>[1,2,3,4,5] [1,0,1,0,1,0,1,1,0,0]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[1,2,3,5,4]=>[1,2,3,5,4]=>[1,2,3,5,4] [1,0,1,0,1,1,0,0,1,0]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[1,2,4,3,5]=>[1,2,4,3,5]=>[1,2,4,3,5] [1,0,1,0,1,1,0,1,0,0]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[1,2,3,5,4]=>[1,2,3,5,4]=>[1,2,3,5,4] [1,0,1,0,1,1,1,0,0,0]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[1,2,5,3,4]=>[1,2,4,5,3]=>[1,2,5,3,4] [1,0,1,1,0,0,1,0,1,0]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[1,3,2,4,5]=>[1,3,2,4,5]=>[1,3,2,4,5] [1,0,1,1,0,0,1,1,0,0]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[1,3,2,5,4]=>[1,3,2,5,4]=>[1,3,2,5,4] [1,0,1,1,0,1,0,0,1,0]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[1,2,4,3,5]=>[1,2,4,3,5]=>[1,2,4,3,5] [1,0,1,1,0,1,0,1,0,0]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[1,2,3,5,4]=>[1,2,3,5,4]=>[1,2,3,5,4] [1,0,1,1,0,1,1,0,0,0]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[1,2,5,3,4]=>[1,2,4,5,3]=>[1,2,5,3,4] [1,0,1,1,1,0,0,0,1,0]=>[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[1,4,2,3,5]=>[1,3,4,2,5]=>[1,4,2,3,5] [1,0,1,1,1,0,0,1,0,0]=>[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[1,3,2,5,4]=>[1,3,2,5,4]=>[1,3,2,5,4] [1,0,1,1,1,0,1,0,0,0]=>[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]=>[1,2,5,3,4]=>[1,2,4,5,3]=>[1,2,5,3,4] [1,0,1,1,1,1,0,0,0,0]=>[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[1,5,2,3,4]=>[1,3,4,5,2]=>[1,5,2,3,4] [1,1,0,0,1,0,1,0,1,0]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[2,1,3,4,5]=>[2,1,3,4,5]=>[2,1,3,4,5] [1,1,0,0,1,0,1,1,0,0]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[2,1,3,5,4]=>[2,1,3,5,4]=>[2,1,3,5,4] [1,1,0,0,1,1,0,0,1,0]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[2,1,4,3,5]=>[2,1,4,3,5]=>[2,1,4,3,5] [1,1,0,0,1,1,0,1,0,0]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[2,1,3,5,4]=>[2,1,3,5,4]=>[2,1,3,5,4] [1,1,0,0,1,1,1,0,0,0]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[2,1,5,3,4]=>[2,1,4,5,3]=>[2,1,5,3,4] [1,1,0,1,0,0,1,0,1,0]=>[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[1,3,2,4,5]=>[1,3,2,4,5]=>[1,3,2,4,5] [1,1,0,1,0,0,1,1,0,0]=>[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[1,3,2,5,4]=>[1,3,2,5,4]=>[1,3,2,5,4] [1,1,0,1,0,1,0,0,1,0]=>[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[1,2,4,3,5]=>[1,2,4,3,5]=>[1,2,4,3,5] [1,1,0,1,0,1,0,1,0,0]=>[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[1,2,3,5,4]=>[1,2,3,5,4]=>[1,2,3,5,4] [1,1,0,1,0,1,1,0,0,0]=>[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[1,2,5,3,4]=>[1,2,4,5,3]=>[1,2,5,3,4] [1,1,0,1,1,0,0,0,1,0]=>[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[1,4,2,3,5]=>[1,3,4,2,5]=>[1,4,2,3,5] [1,1,0,1,1,0,0,1,0,0]=>[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[1,3,2,5,4]=>[1,3,2,5,4]=>[1,3,2,5,4] [1,1,0,1,1,0,1,0,0,0]=>[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]=>[1,2,5,3,4]=>[1,2,4,5,3]=>[1,2,5,3,4] [1,1,0,1,1,1,0,0,0,0]=>[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[1,5,2,3,4]=>[1,3,4,5,2]=>[1,5,2,3,4] [1,1,1,0,0,0,1,0,1,0]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[3,1,2,4,5]=>[2,3,1,4,5]=>[3,1,2,4,5] [1,1,1,0,0,0,1,1,0,0]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[3,1,2,5,4]=>[2,3,1,5,4]=>[3,1,2,5,4] [1,1,1,0,0,1,0,0,1,0]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[2,1,4,3,5]=>[2,1,4,3,5]=>[2,1,4,3,5] [1,1,1,0,0,1,0,1,0,0]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[2,1,3,5,4]=>[2,1,3,5,4]=>[2,1,3,5,4] [1,1,1,0,0,1,1,0,0,0]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[2,1,5,3,4]=>[2,1,4,5,3]=>[2,1,5,3,4] [1,1,1,0,1,0,0,0,1,0]=>[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[1,4,2,3,5]=>[1,3,4,2,5]=>[1,4,2,3,5] [1,1,1,0,1,0,0,1,0,0]=>[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[1,3,2,5,4]=>[1,3,2,5,4]=>[1,3,2,5,4] [1,1,1,0,1,0,1,0,0,0]=>[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]=>[1,2,5,3,4]=>[1,2,4,5,3]=>[1,2,5,3,4] [1,1,1,0,1,1,0,0,0,0]=>[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[1,5,2,3,4]=>[1,3,4,5,2]=>[1,5,2,3,4] [1,1,1,1,0,0,0,0,1,0]=>[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[4,1,2,3,5]=>[2,3,4,1,5]=>[4,1,2,3,5] [1,1,1,1,0,0,0,1,0,0]=>[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[3,1,2,5,4]=>[2,3,1,5,4]=>[3,1,2,5,4] [1,1,1,1,0,0,1,0,0,0]=>[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]=>[2,1,5,3,4]=>[2,1,4,5,3]=>[2,1,5,3,4] [1,1,1,1,0,1,0,0,0,0]=>[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[1,5,2,3,4]=>[1,3,4,5,2]=>[1,5,2,3,4] [1,1,1,1,1,0,0,0,0,0]=>[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[5,1,2,3,4]=>[2,3,4,5,1]=>[5,1,2,3,4] [1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6] [1,0,1,0,1,0,1,0,1,1,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5] [1,0,1,0,1,0,1,1,0,0,1,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,2,3,5,4,6]=>[1,2,3,5,4,6]=>[1,2,3,5,4,6] [1,0,1,0,1,0,1,1,0,1,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5] [1,0,1,0,1,0,1,1,1,0,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,3,6,4,5]=>[1,2,3,5,6,4]=>[1,2,3,6,4,5] [1,0,1,0,1,1,0,0,1,0,1,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[1,2,4,3,5,6]=>[1,2,4,3,5,6]=>[1,2,4,3,5,6] [1,0,1,0,1,1,0,0,1,1,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5] [1,0,1,0,1,1,0,1,0,0,1,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,2,3,5,4,6]=>[1,2,3,5,4,6]=>[1,2,3,5,4,6] [1,0,1,0,1,1,0,1,0,1,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5] [1,0,1,0,1,1,0,1,1,0,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,3,6,4,5]=>[1,2,3,5,6,4]=>[1,2,3,6,4,5] [1,0,1,0,1,1,1,0,0,0,1,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,2,5,3,4,6]=>[1,2,4,5,3,6]=>[1,2,5,3,4,6] [1,0,1,0,1,1,1,0,0,1,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5] [1,0,1,0,1,1,1,0,1,0,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,3,6,4,5]=>[1,2,3,5,6,4]=>[1,2,3,6,4,5] [1,0,1,0,1,1,1,1,0,0,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,6,3,4,5]=>[1,2,4,5,6,3]=>[1,2,6,3,4,5] [1,0,1,1,0,0,1,0,1,0,1,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[1,3,2,4,5,6]=>[1,3,2,4,5,6]=>[1,3,2,4,5,6] [1,0,1,1,0,0,1,0,1,1,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5] [1,0,1,1,0,0,1,1,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,3,2,5,4,6]=>[1,3,2,5,4,6]=>[1,3,2,5,4,6] [1,0,1,1,0,0,1,1,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5] [1,0,1,1,0,0,1,1,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,3,2,6,4,5]=>[1,3,2,5,6,4]=>[1,3,2,6,4,5] [1,0,1,1,0,1,0,0,1,0,1,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[1,2,4,3,5,6]=>[1,2,4,3,5,6]=>[1,2,4,3,5,6] [1,0,1,1,0,1,0,0,1,1,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5] [1,0,1,1,0,1,0,1,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,2,3,5,4,6]=>[1,2,3,5,4,6]=>[1,2,3,5,4,6] [1,0,1,1,0,1,0,1,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5] [1,0,1,1,0,1,0,1,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,3,6,4,5]=>[1,2,3,5,6,4]=>[1,2,3,6,4,5] [1,0,1,1,0,1,1,0,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,2,5,3,4,6]=>[1,2,4,5,3,6]=>[1,2,5,3,4,6] [1,0,1,1,0,1,1,0,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5] [1,0,1,1,0,1,1,0,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,3,6,4,5]=>[1,2,3,5,6,4]=>[1,2,3,6,4,5] [1,0,1,1,0,1,1,1,0,0,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,6,3,4,5]=>[1,2,4,5,6,3]=>[1,2,6,3,4,5] [1,0,1,1,1,0,0,0,1,0,1,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[1,4,2,3,5,6]=>[1,3,4,2,5,6]=>[1,4,2,3,5,6] [1,0,1,1,1,0,0,0,1,1,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[1,4,2,3,6,5]=>[1,3,4,2,6,5]=>[1,4,2,3,6,5] [1,0,1,1,1,0,0,1,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,3,2,5,4,6]=>[1,3,2,5,4,6]=>[1,3,2,5,4,6] [1,0,1,1,1,0,0,1,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5] [1,0,1,1,1,0,0,1,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,3,2,6,4,5]=>[1,3,2,5,6,4]=>[1,3,2,6,4,5] [1,0,1,1,1,0,1,0,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,2,5,3,4,6]=>[1,2,4,5,3,6]=>[1,2,5,3,4,6] [1,0,1,1,1,0,1,0,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5] [1,0,1,1,1,0,1,0,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,3,6,4,5]=>[1,2,3,5,6,4]=>[1,2,3,6,4,5] [1,0,1,1,1,0,1,1,0,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,6,3,4,5]=>[1,2,4,5,6,3]=>[1,2,6,3,4,5] [1,0,1,1,1,1,0,0,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,5,2,3,4,6]=>[1,3,4,5,2,6]=>[1,5,2,3,4,6] [1,0,1,1,1,1,0,0,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,4,2,3,6,5]=>[1,3,4,2,6,5]=>[1,4,2,3,6,5] [1,0,1,1,1,1,0,0,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,3,2,6,4,5]=>[1,3,2,5,6,4]=>[1,3,2,6,4,5] [1,0,1,1,1,1,0,1,0,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,6,3,4,5]=>[1,2,4,5,6,3]=>[1,2,6,3,4,5] [1,0,1,1,1,1,1,0,0,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,6,2,3,4,5]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5] [1,1,0,0,1,0,1,0,1,0,1,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6] [1,1,0,0,1,0,1,0,1,1,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[2,1,3,4,6,5]=>[2,1,3,4,6,5]=>[2,1,3,4,6,5] [1,1,0,0,1,0,1,1,0,0,1,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[2,1,3,5,4,6]=>[2,1,3,5,4,6]=>[2,1,3,5,4,6] [1,1,0,0,1,0,1,1,0,1,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[2,1,3,4,6,5]=>[2,1,3,4,6,5]=>[2,1,3,4,6,5] [1,1,0,0,1,0,1,1,1,0,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[2,1,3,6,4,5]=>[2,1,3,5,6,4]=>[2,1,3,6,4,5] [1,1,0,0,1,1,0,0,1,0,1,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[2,1,4,3,5,6]=>[2,1,4,3,5,6]=>[2,1,4,3,5,6] [1,1,0,0,1,1,0,0,1,1,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5] [1,1,0,0,1,1,0,1,0,0,1,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[2,1,3,5,4,6]=>[2,1,3,5,4,6]=>[2,1,3,5,4,6] [1,1,0,0,1,1,0,1,0,1,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[2,1,3,4,6,5]=>[2,1,3,4,6,5]=>[2,1,3,4,6,5] [1,1,0,0,1,1,0,1,1,0,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[2,1,3,6,4,5]=>[2,1,3,5,6,4]=>[2,1,3,6,4,5] [1,1,0,0,1,1,1,0,0,0,1,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[2,1,5,3,4,6]=>[2,1,4,5,3,6]=>[2,1,5,3,4,6] [1,1,0,0,1,1,1,0,0,1,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5] [1,1,0,0,1,1,1,0,1,0,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[2,1,3,6,4,5]=>[2,1,3,5,6,4]=>[2,1,3,6,4,5] [1,1,0,0,1,1,1,1,0,0,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[2,1,6,3,4,5]=>[2,1,4,5,6,3]=>[2,1,6,3,4,5] [1,1,0,1,0,0,1,0,1,0,1,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[1,3,2,4,5,6]=>[1,3,2,4,5,6]=>[1,3,2,4,5,6] [1,1,0,1,0,0,1,0,1,1,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5] [1,1,0,1,0,0,1,1,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,3,2,5,4,6]=>[1,3,2,5,4,6]=>[1,3,2,5,4,6] [1,1,0,1,0,0,1,1,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5] [1,1,0,1,0,0,1,1,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,3,2,6,4,5]=>[1,3,2,5,6,4]=>[1,3,2,6,4,5] [1,1,0,1,0,1,0,0,1,0,1,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[1,2,4,3,5,6]=>[1,2,4,3,5,6]=>[1,2,4,3,5,6] [1,1,0,1,0,1,0,0,1,1,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5] [1,1,0,1,0,1,0,1,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,2,3,5,4,6]=>[1,2,3,5,4,6]=>[1,2,3,5,4,6] [1,1,0,1,0,1,0,1,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5]=>[1,2,3,4,6,5] [1,1,0,1,0,1,0,1,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,3,6,4,5]=>[1,2,3,5,6,4]=>[1,2,3,6,4,5] [1,1,0,1,0,1,1,0,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,2,5,3,4,6]=>[1,2,4,5,3,6]=>[1,2,5,3,4,6] [1,1,0,1,0,1,1,0,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5] [1,1,0,1,0,1,1,0,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,3,6,4,5]=>[1,2,3,5,6,4]=>[1,2,3,6,4,5] [1,1,0,1,0,1,1,1,0,0,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,6,3,4,5]=>[1,2,4,5,6,3]=>[1,2,6,3,4,5] [1,1,0,1,1,0,0,0,1,0,1,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[1,4,2,3,5,6]=>[1,3,4,2,5,6]=>[1,4,2,3,5,6] [1,1,0,1,1,0,0,0,1,1,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[1,4,2,3,6,5]=>[1,3,4,2,6,5]=>[1,4,2,3,6,5] [1,1,0,1,1,0,0,1,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,3,2,5,4,6]=>[1,3,2,5,4,6]=>[1,3,2,5,4,6] [1,1,0,1,1,0,0,1,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5] [1,1,0,1,1,0,0,1,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,3,2,6,4,5]=>[1,3,2,5,6,4]=>[1,3,2,6,4,5] [1,1,0,1,1,0,1,0,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,2,5,3,4,6]=>[1,2,4,5,3,6]=>[1,2,5,3,4,6] [1,1,0,1,1,0,1,0,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5] [1,1,0,1,1,0,1,0,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,3,6,4,5]=>[1,2,3,5,6,4]=>[1,2,3,6,4,5] [1,1,0,1,1,0,1,1,0,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,6,3,4,5]=>[1,2,4,5,6,3]=>[1,2,6,3,4,5] [1,1,0,1,1,1,0,0,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,5,2,3,4,6]=>[1,3,4,5,2,6]=>[1,5,2,3,4,6] [1,1,0,1,1,1,0,0,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,4,2,3,6,5]=>[1,3,4,2,6,5]=>[1,4,2,3,6,5] [1,1,0,1,1,1,0,0,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,3,2,6,4,5]=>[1,3,2,5,6,4]=>[1,3,2,6,4,5] [1,1,0,1,1,1,0,1,0,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,6,3,4,5]=>[1,2,4,5,6,3]=>[1,2,6,3,4,5] [1,1,0,1,1,1,1,0,0,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,6,2,3,4,5]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5] [1,1,1,0,0,0,1,0,1,0,1,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[3,1,2,4,5,6]=>[2,3,1,4,5,6]=>[3,1,2,4,5,6] [1,1,1,0,0,0,1,0,1,1,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[3,1,2,4,6,5]=>[2,3,1,4,6,5]=>[3,1,2,4,6,5] [1,1,1,0,0,0,1,1,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[3,1,2,5,4,6]=>[2,3,1,5,4,6]=>[3,1,2,5,4,6] [1,1,1,0,0,0,1,1,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[3,1,2,4,6,5]=>[2,3,1,4,6,5]=>[3,1,2,4,6,5] [1,1,1,0,0,0,1,1,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[3,1,2,6,4,5]=>[2,3,1,5,6,4]=>[3,1,2,6,4,5] [1,1,1,0,0,1,0,0,1,0,1,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[2,1,4,3,5,6]=>[2,1,4,3,5,6]=>[2,1,4,3,5,6] [1,1,1,0,0,1,0,0,1,1,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5] [1,1,1,0,0,1,0,1,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[2,1,3,5,4,6]=>[2,1,3,5,4,6]=>[2,1,3,5,4,6] [1,1,1,0,0,1,0,1,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[2,1,3,4,6,5]=>[2,1,3,4,6,5]=>[2,1,3,4,6,5] [1,1,1,0,0,1,0,1,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[2,1,3,6,4,5]=>[2,1,3,5,6,4]=>[2,1,3,6,4,5] [1,1,1,0,0,1,1,0,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[2,1,5,3,4,6]=>[2,1,4,5,3,6]=>[2,1,5,3,4,6] [1,1,1,0,0,1,1,0,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5] [1,1,1,0,0,1,1,0,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[2,1,3,6,4,5]=>[2,1,3,5,6,4]=>[2,1,3,6,4,5] [1,1,1,0,0,1,1,1,0,0,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[2,1,6,3,4,5]=>[2,1,4,5,6,3]=>[2,1,6,3,4,5] [1,1,1,0,1,0,0,0,1,0,1,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[1,4,2,3,5,6]=>[1,3,4,2,5,6]=>[1,4,2,3,5,6] [1,1,1,0,1,0,0,0,1,1,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[1,4,2,3,6,5]=>[1,3,4,2,6,5]=>[1,4,2,3,6,5] [1,1,1,0,1,0,0,1,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,3,2,5,4,6]=>[1,3,2,5,4,6]=>[1,3,2,5,4,6] [1,1,1,0,1,0,0,1,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5]=>[1,3,2,4,6,5] [1,1,1,0,1,0,0,1,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,3,2,6,4,5]=>[1,3,2,5,6,4]=>[1,3,2,6,4,5] [1,1,1,0,1,0,1,0,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,2,5,3,4,6]=>[1,2,4,5,3,6]=>[1,2,5,3,4,6] [1,1,1,0,1,0,1,0,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5]=>[1,2,4,3,6,5] [1,1,1,0,1,0,1,0,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,3,6,4,5]=>[1,2,3,5,6,4]=>[1,2,3,6,4,5] [1,1,1,0,1,0,1,1,0,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,6,3,4,5]=>[1,2,4,5,6,3]=>[1,2,6,3,4,5] [1,1,1,0,1,1,0,0,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,5,2,3,4,6]=>[1,3,4,5,2,6]=>[1,5,2,3,4,6] [1,1,1,0,1,1,0,0,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,4,2,3,6,5]=>[1,3,4,2,6,5]=>[1,4,2,3,6,5] [1,1,1,0,1,1,0,0,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,3,2,6,4,5]=>[1,3,2,5,6,4]=>[1,3,2,6,4,5] [1,1,1,0,1,1,0,1,0,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,6,3,4,5]=>[1,2,4,5,6,3]=>[1,2,6,3,4,5] [1,1,1,0,1,1,1,0,0,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,6,2,3,4,5]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5] [1,1,1,1,0,0,0,0,1,0,1,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[4,1,2,3,5,6]=>[2,3,4,1,5,6]=>[4,1,2,3,5,6] [1,1,1,1,0,0,0,0,1,1,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[4,1,2,3,6,5]=>[2,3,4,1,6,5]=>[4,1,2,3,6,5] [1,1,1,1,0,0,0,1,0,0,1,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[3,1,2,5,4,6]=>[2,3,1,5,4,6]=>[3,1,2,5,4,6] [1,1,1,1,0,0,0,1,0,1,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[3,1,2,4,6,5]=>[2,3,1,4,6,5]=>[3,1,2,4,6,5] [1,1,1,1,0,0,0,1,1,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[3,1,2,6,4,5]=>[2,3,1,5,6,4]=>[3,1,2,6,4,5] [1,1,1,1,0,0,1,0,0,0,1,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[2,1,5,3,4,6]=>[2,1,4,5,3,6]=>[2,1,5,3,4,6] [1,1,1,1,0,0,1,0,0,1,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5]=>[2,1,4,3,6,5] [1,1,1,1,0,0,1,0,1,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[2,1,3,6,4,5]=>[2,1,3,5,6,4]=>[2,1,3,6,4,5] [1,1,1,1,0,0,1,1,0,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[2,1,6,3,4,5]=>[2,1,4,5,6,3]=>[2,1,6,3,4,5] [1,1,1,1,0,1,0,0,0,0,1,0]=>[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[1,5,2,3,4,6]=>[1,3,4,5,2,6]=>[1,5,2,3,4,6] [1,1,1,1,0,1,0,0,0,1,0,0]=>[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[1,4,2,3,6,5]=>[1,3,4,2,6,5]=>[1,4,2,3,6,5] [1,1,1,1,0,1,0,0,1,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,3,2,6,4,5]=>[1,3,2,5,6,4]=>[1,3,2,6,4,5] [1,1,1,1,0,1,0,1,0,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,2,6,3,4,5]=>[1,2,4,5,6,3]=>[1,2,6,3,4,5] [1,1,1,1,0,1,1,0,0,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,6,2,3,4,5]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5] [1,1,1,1,1,0,0,0,0,0,1,0]=>[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[5,1,2,3,4,6]=>[2,3,4,5,1,6]=>[5,1,2,3,4,6] [1,1,1,1,1,0,0,0,0,1,0,0]=>[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[4,1,2,3,6,5]=>[2,3,4,1,6,5]=>[4,1,2,3,6,5] [1,1,1,1,1,0,0,0,1,0,0,0]=>[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[3,1,2,6,4,5]=>[2,3,1,5,6,4]=>[3,1,2,6,4,5] [1,1,1,1,1,0,0,1,0,0,0,0]=>[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[2,1,6,3,4,5]=>[2,1,4,5,6,3]=>[2,1,6,3,4,5] [1,1,1,1,1,0,1,0,0,0,0,0]=>[[0,0,0,0,1,0],[1,0,0,0,-1,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[1,6,2,3,4,5]=>[1,3,4,5,6,2]=>[1,6,2,3,4,5] [1,1,1,1,1,1,0,0,0,0,0,0]=>[[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[6,1,2,3,4,5]=>[2,3,4,5,6,1]=>[6,1,2,3,4,5]
Map
to alternating sign matrix
Description
Return the Dyck path as an alternating sign matrix.
This is an inclusion map from Dyck words of length $2n$ to certain
$n \times n$ alternating sign matrices.
Map
to left key permutation
Description
Return the permutation of the left key of an alternating sign matrix.
This was originally defined by Lascoux and then further studied by Aval [1].
Map
inverse
Description
Sends a permutation to its inverse.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.