Identifier
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Images
=>
Cc0005;cc-rep-0Cc0017;cc-rep-1Cc0005;cc-rep-3
[1,0]=>[[1]]=>[1]=>[1,0] [1,0,1,0]=>[[1,0],[0,1]]=>[1,2]=>[1,0,1,0] [1,1,0,0]=>[[0,1],[1,0]]=>[2,1]=>[1,1,0,0] [1,0,1,0,1,0]=>[[1,0,0],[0,1,0],[0,0,1]]=>[1,2,3]=>[1,0,1,0,1,0] [1,0,1,1,0,0]=>[[1,0,0],[0,0,1],[0,1,0]]=>[1,3,2]=>[1,0,1,1,0,0] [1,1,0,0,1,0]=>[[0,1,0],[1,0,0],[0,0,1]]=>[2,1,3]=>[1,1,0,0,1,0] [1,1,0,1,0,0]=>[[0,1,0],[1,-1,1],[0,1,0]]=>[1,3,2]=>[1,0,1,1,0,0] [1,1,1,0,0,0]=>[[0,0,1],[1,0,0],[0,1,0]]=>[3,1,2]=>[1,1,1,0,0,0] [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]=>[1,0,1,0,1,0,1,0] [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]=>[1,0,1,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,0] [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]=>[1,0,1,0,1,1,0,0] [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]=>[1,0,1,1,1,0,0,0] [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]=>[1,1,0,0,1,0,1,0] [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]=>[1,1,0,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,0] [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]=>[1,0,1,0,1,1,0,0] [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]=>[1,0,1,1,1,0,0,0] [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]=>[1,1,1,0,0,0,1,0] [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]=>[1,1,0,0,1,1,0,0] [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]=>[1,0,1,1,1,0,0,0] [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]=>[1,1,1,1,0,0,0,0] [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]=>[1,0,1,0,1,0,1,0,1,0] [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]=>[1,0,1,0,1,0,1,1,0,0] [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]=>[1,0,1,0,1,1,0,0,1,0] [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]=>[1,0,1,0,1,0,1,1,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,1,0,0,1,0,1,0] [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]=>[1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,1,0,0,1,0] [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]=>[1,0,1,0,1,0,1,1,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,1,1,0,0,0,1,0] [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]=>[1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,1,1,1,0,0,0,0] [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]=>[1,1,0,0,1,0,1,0,1,0] [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]=>[1,1,0,0,1,0,1,1,0,0] [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]=>[1,1,0,0,1,1,0,0,1,0] [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]=>[1,1,0,0,1,0,1,1,0,0] [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]=>[1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,1,0,0,1,0,1,0] [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]=>[1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,1,0,0,1,0] [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]=>[1,0,1,0,1,0,1,1,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,1,1,0,0,0,1,0] [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]=>[1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,1,1,1,0,0,0,0] [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]=>[1,1,1,0,0,0,1,0,1,0] [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]=>[1,1,1,0,0,0,1,1,0,0] [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]=>[1,1,0,0,1,1,0,0,1,0] [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]=>[1,1,0,0,1,0,1,1,0,0] [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]=>[1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,1,1,0,0,0,1,0] [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]=>[1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,1,1,1,0,0,0,0] [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]=>[1,1,1,1,0,0,0,0,1,0] [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]=>[1,1,1,0,0,0,1,1,0,0] [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]=>[1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,1,1,1,0,0,0,0] [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]=>[1,1,1,1,1,0,0,0,0,0] [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]=>[1,0,1,0,1,0,1,0,1,0,1,0] [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]=>[1,0,1,0,1,0,1,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,0,0,1,0] [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]=>[1,0,1,0,1,0,1,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,0,0,1,0,1,0] [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]=>[1,0,1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,0,0,1,0] [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]=>[1,0,1,0,1,0,1,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0,1,0] [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]=>[1,0,1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,0,0,1,0,1,0,1,0] [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]=>[1,0,1,1,0,0,1,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,0,0,1,0] [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]=>[1,0,1,1,0,0,1,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,0,0,1,0,1,0] [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]=>[1,0,1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,0,0,1,0] [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]=>[1,0,1,0,1,0,1,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0,1,0] [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]=>[1,0,1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,0,0,0,1,0,1,0] [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]=>[1,0,1,1,1,0,0,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,0,0,1,0] [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]=>[1,0,1,1,0,0,1,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0,1,0] [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]=>[1,0,1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,1,0,0,0,0,1,0] [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]=>[1,0,1,1,1,0,0,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,1,1,0,0,0,0,0] [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]=>[1,1,0,0,1,0,1,0,1,0,1,0] [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]=>[1,1,0,0,1,0,1,0,1,1,0,0] [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]=>[1,1,0,0,1,0,1,1,0,0,1,0] [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]=>[1,1,0,0,1,0,1,0,1,1,0,0] [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]=>[1,1,0,0,1,0,1,1,1,0,0,0] [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]=>[1,1,0,0,1,1,0,0,1,0,1,0] [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]=>[1,1,0,0,1,1,0,0,1,1,0,0] [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]=>[1,1,0,0,1,0,1,1,0,0,1,0] [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]=>[1,1,0,0,1,0,1,0,1,1,0,0] [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]=>[1,1,0,0,1,0,1,1,1,0,0,0] [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]=>[1,1,0,0,1,1,1,0,0,0,1,0] [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]=>[1,1,0,0,1,1,0,0,1,1,0,0] [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]=>[1,1,0,0,1,0,1,1,1,0,0,0] [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]=>[1,1,0,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,0,0,1,0,1,0,1,0] [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]=>[1,0,1,1,0,0,1,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,0,0,1,0] [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]=>[1,0,1,1,0,0,1,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,0,0,1,0,1,0] [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]=>[1,0,1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,0,0,1,0] [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]=>[1,0,1,0,1,0,1,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0,1,0] [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]=>[1,0,1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,0,0,0,1,0,1,0] [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]=>[1,0,1,1,1,0,0,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,0,0,1,0] [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]=>[1,0,1,1,0,0,1,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0,1,0] [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]=>[1,0,1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,1,0,0,0,0,1,0] [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]=>[1,0,1,1,1,0,0,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,1,1,0,0,0,0,0] [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]=>[1,1,1,0,0,0,1,0,1,0,1,0] [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]=>[1,1,1,0,0,0,1,0,1,1,0,0] [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]=>[1,1,1,0,0,0,1,1,0,0,1,0] [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]=>[1,1,1,0,0,0,1,0,1,1,0,0] [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]=>[1,1,1,0,0,0,1,1,1,0,0,0] [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]=>[1,1,0,0,1,1,0,0,1,0,1,0] [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]=>[1,1,0,0,1,1,0,0,1,1,0,0] [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]=>[1,1,0,0,1,0,1,1,0,0,1,0] [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]=>[1,1,0,0,1,0,1,0,1,1,0,0] [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]=>[1,1,0,0,1,0,1,1,1,0,0,0] [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]=>[1,1,0,0,1,1,1,0,0,0,1,0] [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]=>[1,1,0,0,1,1,0,0,1,1,0,0] [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]=>[1,1,0,0,1,0,1,1,1,0,0,0] [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]=>[1,1,0,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,0,0,0,1,0,1,0] [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]=>[1,0,1,1,1,0,0,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,0,0,1,0] [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]=>[1,0,1,1,0,0,1,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,0,0,0,1,0] [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]=>[1,0,1,0,1,1,0,0,1,1,0,0] [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]=>[1,0,1,0,1,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,1,0,0,0,0,1,0] [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]=>[1,0,1,1,1,0,0,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,1,1,0,0,0,0,0] [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]=>[1,1,1,1,0,0,0,0,1,0,1,0] [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]=>[1,1,1,1,0,0,0,0,1,1,0,0] [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]=>[1,1,1,0,0,0,1,1,0,0,1,0] [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]=>[1,1,1,0,0,0,1,0,1,1,0,0] [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]=>[1,1,1,0,0,0,1,1,1,0,0,0] [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]=>[1,1,0,0,1,1,1,0,0,0,1,0] [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]=>[1,1,0,0,1,1,0,0,1,1,0,0] [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]=>[1,1,0,0,1,0,1,1,1,0,0,0] [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]=>[1,1,0,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,1,0,0,0,0,1,0] [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]=>[1,0,1,1,1,0,0,0,1,1,0,0] [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]=>[1,0,1,1,0,0,1,1,1,0,0,0] [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]=>[1,0,1,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,1,1,0,0,0,0,0] [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]=>[1,1,1,1,1,0,0,0,0,0,1,0] [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]=>[1,1,1,1,0,0,0,0,1,1,0,0] [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]=>[1,1,1,0,0,0,1,1,1,0,0,0] [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]=>[1,1,0,0,1,1,1,1,0,0,0,0] [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]=>[1,0,1,1,1,1,1,0,0,0,0,0] [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]=>[1,1,1,1,1,1,0,0,0,0,0,0]
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
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.