Identifier
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
=>
Cc0007;cc-rep-0
[[1]]=>[1]=>[1]=>[1]=>
[[1,2]]=>[1,2]=>[1,2]=>[1,2]=>1
[[1],[2]]=>[2,1]=>[2,1]=>[2,1]=>0
[[1,2,3]]=>[1,2,3]=>[1,2,3]=>[1,2,3]=>11
[[1,3],[2]]=>[2,1,3]=>[2,1,3]=>[2,1,3]=>01
[[1,2],[3]]=>[3,1,2]=>[2,3,1]=>[2,3,1]=>00
[[1],[2],[3]]=>[3,2,1]=>[3,2,1]=>[3,2,1]=>00
[[1,2,3,4]]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3,4]=>111
[[1,3,4],[2]]=>[2,1,3,4]=>[2,1,3,4]=>[2,1,3,4]=>011
[[1,2,4],[3]]=>[3,1,2,4]=>[2,3,1,4]=>[2,3,1,4]=>001
[[1,2,3],[4]]=>[4,1,2,3]=>[2,3,4,1]=>[2,3,4,1]=>000
[[1,3],[2,4]]=>[2,4,1,3]=>[1,3,4,2]=>[3,1,4,2]=>000
[[1,2],[3,4]]=>[3,4,1,2]=>[3,1,4,2]=>[3,4,1,2]=>000
[[1,4],[2],[3]]=>[3,2,1,4]=>[3,2,1,4]=>[3,2,1,4]=>001
[[1,3],[2],[4]]=>[4,2,1,3]=>[3,2,4,1]=>[3,2,4,1]=>000
[[1,2],[3],[4]]=>[4,3,1,2]=>[3,4,2,1]=>[3,4,2,1]=>000
[[1],[2],[3],[4]]=>[4,3,2,1]=>[4,3,2,1]=>[4,3,2,1]=>000
[[1,2,3,4,5]]=>[1,2,3,4,5]=>[1,2,3,4,5]=>[1,2,3,4,5]=>1111
[[1,3,4,5],[2]]=>[2,1,3,4,5]=>[2,1,3,4,5]=>[2,1,3,4,5]=>0111
[[1,2,4,5],[3]]=>[3,1,2,4,5]=>[2,3,1,4,5]=>[2,3,1,4,5]=>0011
[[1,2,3,5],[4]]=>[4,1,2,3,5]=>[2,3,4,1,5]=>[2,3,4,1,5]=>0001
[[1,2,3,4],[5]]=>[5,1,2,3,4]=>[2,3,4,5,1]=>[2,3,4,5,1]=>0000
[[1,3,5],[2,4]]=>[2,4,1,3,5]=>[1,3,4,2,5]=>[3,1,4,2,5]=>0001
[[1,2,5],[3,4]]=>[3,4,1,2,5]=>[3,1,4,2,5]=>[3,4,1,2,5]=>0001
[[1,3,4],[2,5]]=>[2,5,1,3,4]=>[1,3,4,5,2]=>[3,1,4,5,2]=>0000
[[1,2,4],[3,5]]=>[3,5,1,2,4]=>[3,1,4,5,2]=>[3,4,1,5,2]=>0000
[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[3,4,1,5,2]=>[3,4,5,1,2]=>0000
[[1,4,5],[2],[3]]=>[3,2,1,4,5]=>[3,2,1,4,5]=>[3,2,1,4,5]=>0011
[[1,3,5],[2],[4]]=>[4,2,1,3,5]=>[3,2,4,1,5]=>[3,2,4,1,5]=>0001
[[1,2,5],[3],[4]]=>[4,3,1,2,5]=>[3,4,2,1,5]=>[3,4,2,1,5]=>0001
[[1,3,4],[2],[5]]=>[5,2,1,3,4]=>[3,2,4,5,1]=>[3,2,4,5,1]=>0000
[[1,2,4],[3],[5]]=>[5,3,1,2,4]=>[3,4,2,5,1]=>[3,4,2,5,1]=>0000
[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[3,4,5,2,1]=>[3,4,5,2,1]=>0000
[[1,4],[2,5],[3]]=>[3,2,5,1,4]=>[2,1,4,5,3]=>[4,2,1,5,3]=>0000
[[1,3],[2,5],[4]]=>[4,2,5,1,3]=>[2,4,1,5,3]=>[4,2,5,1,3]=>0000
[[1,2],[3,5],[4]]=>[4,3,5,1,2]=>[4,2,1,5,3]=>[4,5,2,1,3]=>0000
[[1,3],[2,4],[5]]=>[5,2,4,1,3]=>[2,4,5,3,1]=>[4,2,5,3,1]=>0000
[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[4,2,5,3,1]=>[4,5,2,3,1]=>0000
[[1,5],[2],[3],[4]]=>[4,3,2,1,5]=>[4,3,2,1,5]=>[4,3,2,1,5]=>0001
[[1,4],[2],[3],[5]]=>[5,3,2,1,4]=>[4,3,2,5,1]=>[4,3,2,5,1]=>0000
[[1,3],[2],[4],[5]]=>[5,4,2,1,3]=>[4,3,5,2,1]=>[4,3,5,2,1]=>0000
[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[4,5,3,2,1]=>[4,5,3,2,1]=>0000
[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[5,4,3,2,1]=>[5,4,3,2,1]=>0000
[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>11111
[[1,3,4,5,6],[2]]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>01111
[[1,2,4,5,6],[3]]=>[3,1,2,4,5,6]=>[2,3,1,4,5,6]=>[2,3,1,4,5,6]=>00111
[[1,2,3,5,6],[4]]=>[4,1,2,3,5,6]=>[2,3,4,1,5,6]=>[2,3,4,1,5,6]=>00011
[[1,2,3,4,6],[5]]=>[5,1,2,3,4,6]=>[2,3,4,5,1,6]=>[2,3,4,5,1,6]=>00001
[[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>00000
[[1,3,5,6],[2,4]]=>[2,4,1,3,5,6]=>[1,3,4,2,5,6]=>[3,1,4,2,5,6]=>00011
[[1,2,5,6],[3,4]]=>[3,4,1,2,5,6]=>[3,1,4,2,5,6]=>[3,4,1,2,5,6]=>00011
[[1,3,4,6],[2,5]]=>[2,5,1,3,4,6]=>[1,3,4,5,2,6]=>[3,1,4,5,2,6]=>00001
[[1,2,4,6],[3,5]]=>[3,5,1,2,4,6]=>[3,1,4,5,2,6]=>[3,4,1,5,2,6]=>00001
[[1,2,3,6],[4,5]]=>[4,5,1,2,3,6]=>[3,4,1,5,2,6]=>[3,4,5,1,2,6]=>00001
[[1,3,4,5],[2,6]]=>[2,6,1,3,4,5]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>00000
[[1,2,4,5],[3,6]]=>[3,6,1,2,4,5]=>[3,1,4,5,6,2]=>[3,4,1,5,6,2]=>00000
[[1,2,3,5],[4,6]]=>[4,6,1,2,3,5]=>[3,4,1,5,6,2]=>[3,4,5,1,6,2]=>00000
[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[3,4,5,1,6,2]=>[3,4,5,6,1,2]=>00000
[[1,4,5,6],[2],[3]]=>[3,2,1,4,5,6]=>[3,2,1,4,5,6]=>[3,2,1,4,5,6]=>00111
[[1,3,5,6],[2],[4]]=>[4,2,1,3,5,6]=>[3,2,4,1,5,6]=>[3,2,4,1,5,6]=>00011
[[1,2,5,6],[3],[4]]=>[4,3,1,2,5,6]=>[3,4,2,1,5,6]=>[3,4,2,1,5,6]=>00011
[[1,3,4,6],[2],[5]]=>[5,2,1,3,4,6]=>[3,2,4,5,1,6]=>[3,2,4,5,1,6]=>00001
[[1,2,4,6],[3],[5]]=>[5,3,1,2,4,6]=>[3,4,2,5,1,6]=>[3,4,2,5,1,6]=>00001
[[1,2,3,6],[4],[5]]=>[5,4,1,2,3,6]=>[3,4,5,2,1,6]=>[3,4,5,2,1,6]=>00001
[[1,3,4,5],[2],[6]]=>[6,2,1,3,4,5]=>[3,2,4,5,6,1]=>[3,2,4,5,6,1]=>00000
[[1,2,4,5],[3],[6]]=>[6,3,1,2,4,5]=>[3,4,2,5,6,1]=>[3,4,2,5,6,1]=>00000
[[1,2,3,5],[4],[6]]=>[6,4,1,2,3,5]=>[3,4,5,2,6,1]=>[3,4,5,2,6,1]=>00000
[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[3,4,5,6,2,1]=>[3,4,5,6,2,1]=>00000
[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[4,1,2,5,6,3]=>[4,1,5,2,6,3]=>00000
[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[1,4,2,5,6,3]=>[4,5,1,2,6,3]=>00000
[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[4,5,1,2,6,3]=>[4,1,5,6,2,3]=>00000
[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[1,4,5,2,6,3]=>[4,5,1,6,2,3]=>00000
[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>[4,5,6,1,2,3]=>00000
[[1,4,6],[2,5],[3]]=>[3,2,5,1,4,6]=>[2,1,4,5,3,6]=>[4,2,1,5,3,6]=>00001
[[1,3,6],[2,5],[4]]=>[4,2,5,1,3,6]=>[2,4,1,5,3,6]=>[4,2,5,1,3,6]=>00001
[[1,2,6],[3,5],[4]]=>[4,3,5,1,2,6]=>[4,2,1,5,3,6]=>[4,5,2,1,3,6]=>00001
[[1,3,6],[2,4],[5]]=>[5,2,4,1,3,6]=>[2,4,5,3,1,6]=>[4,2,5,3,1,6]=>00001
[[1,2,6],[3,4],[5]]=>[5,3,4,1,2,6]=>[4,2,5,3,1,6]=>[4,5,2,3,1,6]=>00001
[[1,4,5],[2,6],[3]]=>[3,2,6,1,4,5]=>[2,1,4,5,6,3]=>[4,2,1,5,6,3]=>00000
[[1,3,5],[2,6],[4]]=>[4,2,6,1,3,5]=>[2,4,1,5,6,3]=>[4,2,5,1,6,3]=>00000
[[1,2,5],[3,6],[4]]=>[4,3,6,1,2,5]=>[4,2,1,5,6,3]=>[4,5,2,1,6,3]=>00000
[[1,3,4],[2,6],[5]]=>[5,2,6,1,3,4]=>[2,4,5,1,6,3]=>[4,2,5,6,1,3]=>00000
[[1,2,4],[3,6],[5]]=>[5,3,6,1,2,4]=>[4,2,5,1,6,3]=>[4,5,2,6,1,3]=>00000
[[1,2,3],[4,6],[5]]=>[5,4,6,1,2,3]=>[4,5,2,1,6,3]=>[4,5,6,2,1,3]=>00000
[[1,3,5],[2,4],[6]]=>[6,2,4,1,3,5]=>[2,4,5,3,6,1]=>[4,2,5,3,6,1]=>00000
[[1,2,5],[3,4],[6]]=>[6,3,4,1,2,5]=>[4,2,5,3,6,1]=>[4,5,2,3,6,1]=>00000
[[1,3,4],[2,5],[6]]=>[6,2,5,1,3,4]=>[2,4,5,6,3,1]=>[4,2,5,6,3,1]=>00000
[[1,2,4],[3,5],[6]]=>[6,3,5,1,2,4]=>[4,2,5,6,3,1]=>[4,5,2,6,3,1]=>00000
[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[4,5,2,6,3,1]=>[4,5,6,2,3,1]=>00000
[[1,5,6],[2],[3],[4]]=>[4,3,2,1,5,6]=>[4,3,2,1,5,6]=>[4,3,2,1,5,6]=>00011
[[1,4,6],[2],[3],[5]]=>[5,3,2,1,4,6]=>[4,3,2,5,1,6]=>[4,3,2,5,1,6]=>00001
[[1,3,6],[2],[4],[5]]=>[5,4,2,1,3,6]=>[4,3,5,2,1,6]=>[4,3,5,2,1,6]=>00001
[[1,2,6],[3],[4],[5]]=>[5,4,3,1,2,6]=>[4,5,3,2,1,6]=>[4,5,3,2,1,6]=>00001
[[1,4,5],[2],[3],[6]]=>[6,3,2,1,4,5]=>[4,3,2,5,6,1]=>[4,3,2,5,6,1]=>00000
[[1,3,5],[2],[4],[6]]=>[6,4,2,1,3,5]=>[4,3,5,2,6,1]=>[4,3,5,2,6,1]=>00000
[[1,2,5],[3],[4],[6]]=>[6,4,3,1,2,5]=>[4,5,3,2,6,1]=>[4,5,3,2,6,1]=>00000
[[1,3,4],[2],[5],[6]]=>[6,5,2,1,3,4]=>[4,3,5,6,2,1]=>[4,3,5,6,2,1]=>00000
[[1,2,4],[3],[5],[6]]=>[6,5,3,1,2,4]=>[4,5,3,6,2,1]=>[4,5,3,6,2,1]=>00000
[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[4,5,6,3,2,1]=>[4,5,6,3,2,1]=>00000
[[1,4],[2,5],[3,6]]=>[3,6,2,5,1,4]=>[1,3,5,6,4,2]=>[5,3,1,6,4,2]=>00000
[[1,3],[2,5],[4,6]]=>[4,6,2,5,1,3]=>[3,1,5,6,4,2]=>[5,3,6,1,4,2]=>00000
[[1,2],[3,5],[4,6]]=>[4,6,3,5,1,2]=>[1,5,3,6,4,2]=>[5,6,3,1,4,2]=>00000
[[1,3],[2,4],[5,6]]=>[5,6,2,4,1,3]=>[3,5,1,6,4,2]=>[5,3,6,4,1,2]=>00000
[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[5,3,1,6,4,2]=>[5,6,3,4,1,2]=>00000
[[1,5],[2,6],[3],[4]]=>[4,3,2,6,1,5]=>[3,2,1,5,6,4]=>[5,3,2,1,6,4]=>00000
[[1,4],[2,6],[3],[5]]=>[5,3,2,6,1,4]=>[3,2,5,1,6,4]=>[5,3,2,6,1,4]=>00000
[[1,3],[2,6],[4],[5]]=>[5,4,2,6,1,3]=>[3,5,2,1,6,4]=>[5,3,6,2,1,4]=>00000
[[1,2],[3,6],[4],[5]]=>[5,4,3,6,1,2]=>[5,3,2,1,6,4]=>[5,6,3,2,1,4]=>00000
[[1,4],[2,5],[3],[6]]=>[6,3,2,5,1,4]=>[3,2,5,6,4,1]=>[5,3,2,6,4,1]=>00000
[[1,3],[2,5],[4],[6]]=>[6,4,2,5,1,3]=>[3,5,2,6,4,1]=>[5,3,6,2,4,1]=>00000
[[1,2],[3,5],[4],[6]]=>[6,4,3,5,1,2]=>[5,3,2,6,4,1]=>[5,6,3,2,4,1]=>00000
[[1,3],[2,4],[5],[6]]=>[6,5,2,4,1,3]=>[3,5,6,4,2,1]=>[5,3,6,4,2,1]=>00000
[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[5,3,6,4,2,1]=>[5,6,3,4,2,1]=>00000
[[1,6],[2],[3],[4],[5]]=>[5,4,3,2,1,6]=>[5,4,3,2,1,6]=>[5,4,3,2,1,6]=>00001
[[1,5],[2],[3],[4],[6]]=>[6,4,3,2,1,5]=>[5,4,3,2,6,1]=>[5,4,3,2,6,1]=>00000
[[1,4],[2],[3],[5],[6]]=>[6,5,3,2,1,4]=>[5,4,3,6,2,1]=>[5,4,3,6,2,1]=>00000
[[1,3],[2],[4],[5],[6]]=>[6,5,4,2,1,3]=>[5,4,6,3,2,1]=>[5,4,6,3,2,1]=>00000
[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[5,6,4,3,2,1]=>[5,6,4,3,2,1]=>00000
[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>00000
[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7]=>111111
[[1,3,4,5,6,7],[2]]=>[2,1,3,4,5,6,7]=>[2,1,3,4,5,6,7]=>[2,1,3,4,5,6,7]=>011111
[[1,2,4,5,6,7],[3]]=>[3,1,2,4,5,6,7]=>[2,3,1,4,5,6,7]=>[2,3,1,4,5,6,7]=>001111
[[1,2,3,5,6,7],[4]]=>[4,1,2,3,5,6,7]=>[2,3,4,1,5,6,7]=>[2,3,4,1,5,6,7]=>000111
[[1,2,3,4,6,7],[5]]=>[5,1,2,3,4,6,7]=>[2,3,4,5,1,6,7]=>[2,3,4,5,1,6,7]=>000011
[[1,2,3,4,5,7],[6]]=>[6,1,2,3,4,5,7]=>[2,3,4,5,6,1,7]=>[2,3,4,5,6,1,7]=>000001
[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[2,3,4,5,6,7,1]=>[2,3,4,5,6,7,1]=>000000
[[1,3,5,6,7],[2,4]]=>[2,4,1,3,5,6,7]=>[1,3,4,2,5,6,7]=>[3,1,4,2,5,6,7]=>000111
[[1,2,5,6,7],[3,4]]=>[3,4,1,2,5,6,7]=>[3,1,4,2,5,6,7]=>[3,4,1,2,5,6,7]=>000111
[[1,3,4,6,7],[2,5]]=>[2,5,1,3,4,6,7]=>[1,3,4,5,2,6,7]=>[3,1,4,5,2,6,7]=>000011
[[1,2,4,6,7],[3,5]]=>[3,5,1,2,4,6,7]=>[3,1,4,5,2,6,7]=>[3,4,1,5,2,6,7]=>000011
[[1,2,3,6,7],[4,5]]=>[4,5,1,2,3,6,7]=>[3,4,1,5,2,6,7]=>[3,4,5,1,2,6,7]=>000011
[[1,3,4,5,7],[2,6]]=>[2,6,1,3,4,5,7]=>[1,3,4,5,6,2,7]=>[3,1,4,5,6,2,7]=>000001
[[1,2,4,5,7],[3,6]]=>[3,6,1,2,4,5,7]=>[3,1,4,5,6,2,7]=>[3,4,1,5,6,2,7]=>000001
[[1,2,3,5,7],[4,6]]=>[4,6,1,2,3,5,7]=>[3,4,1,5,6,2,7]=>[3,4,5,1,6,2,7]=>000001
[[1,2,3,4,7],[5,6]]=>[5,6,1,2,3,4,7]=>[3,4,5,1,6,2,7]=>[3,4,5,6,1,2,7]=>000001
[[1,3,4,5,6],[2,7]]=>[2,7,1,3,4,5,6]=>[1,3,4,5,6,7,2]=>[3,1,4,5,6,7,2]=>000000
[[1,2,4,5,6],[3,7]]=>[3,7,1,2,4,5,6]=>[3,1,4,5,6,7,2]=>[3,4,1,5,6,7,2]=>000000
[[1,2,3,5,6],[4,7]]=>[4,7,1,2,3,5,6]=>[3,4,1,5,6,7,2]=>[3,4,5,1,6,7,2]=>000000
[[1,2,3,4,6],[5,7]]=>[5,7,1,2,3,4,6]=>[3,4,5,1,6,7,2]=>[3,4,5,6,1,7,2]=>000000
[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[3,4,5,6,1,7,2]=>[3,4,5,6,7,1,2]=>000000
[[1,4,5,6,7],[2],[3]]=>[3,2,1,4,5,6,7]=>[3,2,1,4,5,6,7]=>[3,2,1,4,5,6,7]=>001111
[[1,3,5,6,7],[2],[4]]=>[4,2,1,3,5,6,7]=>[3,2,4,1,5,6,7]=>[3,2,4,1,5,6,7]=>000111
[[1,2,5,6,7],[3],[4]]=>[4,3,1,2,5,6,7]=>[3,4,2,1,5,6,7]=>[3,4,2,1,5,6,7]=>000111
[[1,3,4,6,7],[2],[5]]=>[5,2,1,3,4,6,7]=>[3,2,4,5,1,6,7]=>[3,2,4,5,1,6,7]=>000011
[[1,2,4,6,7],[3],[5]]=>[5,3,1,2,4,6,7]=>[3,4,2,5,1,6,7]=>[3,4,2,5,1,6,7]=>000011
[[1,2,3,6,7],[4],[5]]=>[5,4,1,2,3,6,7]=>[3,4,5,2,1,6,7]=>[3,4,5,2,1,6,7]=>000011
[[1,3,4,5,7],[2],[6]]=>[6,2,1,3,4,5,7]=>[3,2,4,5,6,1,7]=>[3,2,4,5,6,1,7]=>000001
[[1,2,4,5,7],[3],[6]]=>[6,3,1,2,4,5,7]=>[3,4,2,5,6,1,7]=>[3,4,2,5,6,1,7]=>000001
[[1,2,3,5,7],[4],[6]]=>[6,4,1,2,3,5,7]=>[3,4,5,2,6,1,7]=>[3,4,5,2,6,1,7]=>000001
[[1,2,3,4,7],[5],[6]]=>[6,5,1,2,3,4,7]=>[3,4,5,6,2,1,7]=>[3,4,5,6,2,1,7]=>000001
[[1,3,4,5,6],[2],[7]]=>[7,2,1,3,4,5,6]=>[3,2,4,5,6,7,1]=>[3,2,4,5,6,7,1]=>000000
[[1,2,4,5,6],[3],[7]]=>[7,3,1,2,4,5,6]=>[3,4,2,5,6,7,1]=>[3,4,2,5,6,7,1]=>000000
[[1,2,3,5,6],[4],[7]]=>[7,4,1,2,3,5,6]=>[3,4,5,2,6,7,1]=>[3,4,5,2,6,7,1]=>000000
[[1,2,3,4,6],[5],[7]]=>[7,5,1,2,3,4,6]=>[3,4,5,6,2,7,1]=>[3,4,5,6,2,7,1]=>000000
[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[3,4,5,6,7,2,1]=>[3,4,5,6,7,2,1]=>000000
[[1,3,5,7],[2,4,6]]=>[2,4,6,1,3,5,7]=>[4,1,2,5,6,3,7]=>[4,1,5,2,6,3,7]=>000001
[[1,2,5,7],[3,4,6]]=>[3,4,6,1,2,5,7]=>[1,4,2,5,6,3,7]=>[4,5,1,2,6,3,7]=>000001
[[1,3,4,7],[2,5,6]]=>[2,5,6,1,3,4,7]=>[4,5,1,2,6,3,7]=>[4,1,5,6,2,3,7]=>000001
[[1,2,4,7],[3,5,6]]=>[3,5,6,1,2,4,7]=>[1,4,5,2,6,3,7]=>[4,5,1,6,2,3,7]=>000001
[[1,2,3,7],[4,5,6]]=>[4,5,6,1,2,3,7]=>[4,1,5,2,6,3,7]=>[4,5,6,1,2,3,7]=>000001
[[1,3,5,6],[2,4,7]]=>[2,4,7,1,3,5,6]=>[4,1,2,5,6,7,3]=>[4,1,5,2,6,7,3]=>000000
[[1,2,5,6],[3,4,7]]=>[3,4,7,1,2,5,6]=>[1,4,2,5,6,7,3]=>[4,5,1,2,6,7,3]=>000000
[[1,3,4,6],[2,5,7]]=>[2,5,7,1,3,4,6]=>[4,5,1,2,6,7,3]=>[4,1,5,6,2,7,3]=>000000
[[1,2,4,6],[3,5,7]]=>[3,5,7,1,2,4,6]=>[1,4,5,2,6,7,3]=>[4,5,1,6,2,7,3]=>000000
[[1,2,3,6],[4,5,7]]=>[4,5,7,1,2,3,6]=>[4,1,5,2,6,7,3]=>[4,5,6,1,2,7,3]=>000000
[[1,3,4,5],[2,6,7]]=>[2,6,7,1,3,4,5]=>[4,5,6,1,2,7,3]=>[4,1,5,6,7,2,3]=>000000
[[1,2,4,5],[3,6,7]]=>[3,6,7,1,2,4,5]=>[1,4,5,6,2,7,3]=>[4,5,1,6,7,2,3]=>000000
[[1,2,3,5],[4,6,7]]=>[4,6,7,1,2,3,5]=>[4,1,5,6,2,7,3]=>[4,5,6,1,7,2,3]=>000000
[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[4,5,1,6,2,7,3]=>[4,5,6,7,1,2,3]=>000000
[[1,4,6,7],[2,5],[3]]=>[3,2,5,1,4,6,7]=>[2,1,4,5,3,6,7]=>[4,2,1,5,3,6,7]=>000011
[[1,3,6,7],[2,5],[4]]=>[4,2,5,1,3,6,7]=>[2,4,1,5,3,6,7]=>[4,2,5,1,3,6,7]=>000011
[[1,2,6,7],[3,5],[4]]=>[4,3,5,1,2,6,7]=>[4,2,1,5,3,6,7]=>[4,5,2,1,3,6,7]=>000011
[[1,3,6,7],[2,4],[5]]=>[5,2,4,1,3,6,7]=>[2,4,5,3,1,6,7]=>[4,2,5,3,1,6,7]=>000011
[[1,2,6,7],[3,4],[5]]=>[5,3,4,1,2,6,7]=>[4,2,5,3,1,6,7]=>[4,5,2,3,1,6,7]=>000011
[[1,4,5,7],[2,6],[3]]=>[3,2,6,1,4,5,7]=>[2,1,4,5,6,3,7]=>[4,2,1,5,6,3,7]=>000001
[[1,3,5,7],[2,6],[4]]=>[4,2,6,1,3,5,7]=>[2,4,1,5,6,3,7]=>[4,2,5,1,6,3,7]=>000001
[[1,2,5,7],[3,6],[4]]=>[4,3,6,1,2,5,7]=>[4,2,1,5,6,3,7]=>[4,5,2,1,6,3,7]=>000001
[[1,3,4,7],[2,6],[5]]=>[5,2,6,1,3,4,7]=>[2,4,5,1,6,3,7]=>[4,2,5,6,1,3,7]=>000001
[[1,2,4,7],[3,6],[5]]=>[5,3,6,1,2,4,7]=>[4,2,5,1,6,3,7]=>[4,5,2,6,1,3,7]=>000001
[[1,2,3,7],[4,6],[5]]=>[5,4,6,1,2,3,7]=>[4,5,2,1,6,3,7]=>[4,5,6,2,1,3,7]=>000001
[[1,3,5,7],[2,4],[6]]=>[6,2,4,1,3,5,7]=>[2,4,5,3,6,1,7]=>[4,2,5,3,6,1,7]=>000001
[[1,2,5,7],[3,4],[6]]=>[6,3,4,1,2,5,7]=>[4,2,5,3,6,1,7]=>[4,5,2,3,6,1,7]=>000001
[[1,3,4,7],[2,5],[6]]=>[6,2,5,1,3,4,7]=>[2,4,5,6,3,1,7]=>[4,2,5,6,3,1,7]=>000001
[[1,2,4,7],[3,5],[6]]=>[6,3,5,1,2,4,7]=>[4,2,5,6,3,1,7]=>[4,5,2,6,3,1,7]=>000001
[[1,2,3,7],[4,5],[6]]=>[6,4,5,1,2,3,7]=>[4,5,2,6,3,1,7]=>[4,5,6,2,3,1,7]=>000001
[[1,4,5,6],[2,7],[3]]=>[3,2,7,1,4,5,6]=>[2,1,4,5,6,7,3]=>[4,2,1,5,6,7,3]=>000000
[[1,3,5,6],[2,7],[4]]=>[4,2,7,1,3,5,6]=>[2,4,1,5,6,7,3]=>[4,2,5,1,6,7,3]=>000000
[[1,2,5,6],[3,7],[4]]=>[4,3,7,1,2,5,6]=>[4,2,1,5,6,7,3]=>[4,5,2,1,6,7,3]=>000000
[[1,3,4,6],[2,7],[5]]=>[5,2,7,1,3,4,6]=>[2,4,5,1,6,7,3]=>[4,2,5,6,1,7,3]=>000000
[[1,2,4,6],[3,7],[5]]=>[5,3,7,1,2,4,6]=>[4,2,5,1,6,7,3]=>[4,5,2,6,1,7,3]=>000000
[[1,2,3,6],[4,7],[5]]=>[5,4,7,1,2,3,6]=>[4,5,2,1,6,7,3]=>[4,5,6,2,1,7,3]=>000000
[[1,3,4,5],[2,7],[6]]=>[6,2,7,1,3,4,5]=>[2,4,5,6,1,7,3]=>[4,2,5,6,7,1,3]=>000000
[[1,2,4,5],[3,7],[6]]=>[6,3,7,1,2,4,5]=>[4,2,5,6,1,7,3]=>[4,5,2,6,7,1,3]=>000000
[[1,2,3,5],[4,7],[6]]=>[6,4,7,1,2,3,5]=>[4,5,2,6,1,7,3]=>[4,5,6,2,7,1,3]=>000000
[[1,2,3,4],[5,7],[6]]=>[6,5,7,1,2,3,4]=>[4,5,6,2,1,7,3]=>[4,5,6,7,2,1,3]=>000000
[[1,3,5,6],[2,4],[7]]=>[7,2,4,1,3,5,6]=>[2,4,5,3,6,7,1]=>[4,2,5,3,6,7,1]=>000000
[[1,2,5,6],[3,4],[7]]=>[7,3,4,1,2,5,6]=>[4,2,5,3,6,7,1]=>[4,5,2,3,6,7,1]=>000000
[[1,3,4,6],[2,5],[7]]=>[7,2,5,1,3,4,6]=>[2,4,5,6,3,7,1]=>[4,2,5,6,3,7,1]=>000000
[[1,2,4,6],[3,5],[7]]=>[7,3,5,1,2,4,6]=>[4,2,5,6,3,7,1]=>[4,5,2,6,3,7,1]=>000000
[[1,2,3,6],[4,5],[7]]=>[7,4,5,1,2,3,6]=>[4,5,2,6,3,7,1]=>[4,5,6,2,3,7,1]=>000000
[[1,3,4,5],[2,6],[7]]=>[7,2,6,1,3,4,5]=>[2,4,5,6,7,3,1]=>[4,2,5,6,7,3,1]=>000000
[[1,2,4,5],[3,6],[7]]=>[7,3,6,1,2,4,5]=>[4,2,5,6,7,3,1]=>[4,5,2,6,7,3,1]=>000000
[[1,2,3,5],[4,6],[7]]=>[7,4,6,1,2,3,5]=>[4,5,2,6,7,3,1]=>[4,5,6,2,7,3,1]=>000000
[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[4,5,6,2,7,3,1]=>[4,5,6,7,2,3,1]=>000000
[[1,5,6,7],[2],[3],[4]]=>[4,3,2,1,5,6,7]=>[4,3,2,1,5,6,7]=>[4,3,2,1,5,6,7]=>000111
[[1,4,6,7],[2],[3],[5]]=>[5,3,2,1,4,6,7]=>[4,3,2,5,1,6,7]=>[4,3,2,5,1,6,7]=>000011
[[1,3,6,7],[2],[4],[5]]=>[5,4,2,1,3,6,7]=>[4,3,5,2,1,6,7]=>[4,3,5,2,1,6,7]=>000011
[[1,2,6,7],[3],[4],[5]]=>[5,4,3,1,2,6,7]=>[4,5,3,2,1,6,7]=>[4,5,3,2,1,6,7]=>000011
[[1,4,5,7],[2],[3],[6]]=>[6,3,2,1,4,5,7]=>[4,3,2,5,6,1,7]=>[4,3,2,5,6,1,7]=>000001
[[1,3,5,7],[2],[4],[6]]=>[6,4,2,1,3,5,7]=>[4,3,5,2,6,1,7]=>[4,3,5,2,6,1,7]=>000001
[[1,2,5,7],[3],[4],[6]]=>[6,4,3,1,2,5,7]=>[4,5,3,2,6,1,7]=>[4,5,3,2,6,1,7]=>000001
[[1,3,4,7],[2],[5],[6]]=>[6,5,2,1,3,4,7]=>[4,3,5,6,2,1,7]=>[4,3,5,6,2,1,7]=>000001
[[1,2,4,7],[3],[5],[6]]=>[6,5,3,1,2,4,7]=>[4,5,3,6,2,1,7]=>[4,5,3,6,2,1,7]=>000001
[[1,2,3,7],[4],[5],[6]]=>[6,5,4,1,2,3,7]=>[4,5,6,3,2,1,7]=>[4,5,6,3,2,1,7]=>000001
[[1,4,5,6],[2],[3],[7]]=>[7,3,2,1,4,5,6]=>[4,3,2,5,6,7,1]=>[4,3,2,5,6,7,1]=>000000
[[1,3,5,6],[2],[4],[7]]=>[7,4,2,1,3,5,6]=>[4,3,5,2,6,7,1]=>[4,3,5,2,6,7,1]=>000000
[[1,2,5,6],[3],[4],[7]]=>[7,4,3,1,2,5,6]=>[4,5,3,2,6,7,1]=>[4,5,3,2,6,7,1]=>000000
[[1,3,4,6],[2],[5],[7]]=>[7,5,2,1,3,4,6]=>[4,3,5,6,2,7,1]=>[4,3,5,6,2,7,1]=>000000
[[1,2,4,6],[3],[5],[7]]=>[7,5,3,1,2,4,6]=>[4,5,3,6,2,7,1]=>[4,5,3,6,2,7,1]=>000000
[[1,2,3,6],[4],[5],[7]]=>[7,5,4,1,2,3,6]=>[4,5,6,3,2,7,1]=>[4,5,6,3,2,7,1]=>000000
[[1,3,4,5],[2],[6],[7]]=>[7,6,2,1,3,4,5]=>[4,3,5,6,7,2,1]=>[4,3,5,6,7,2,1]=>000000
[[1,2,4,5],[3],[6],[7]]=>[7,6,3,1,2,4,5]=>[4,5,3,6,7,2,1]=>[4,5,3,6,7,2,1]=>000000
[[1,2,3,5],[4],[6],[7]]=>[7,6,4,1,2,3,5]=>[4,5,6,3,7,2,1]=>[4,5,6,3,7,2,1]=>000000
[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[4,5,6,7,3,2,1]=>[4,5,6,7,3,2,1]=>000000
[[1,4,6],[2,5,7],[3]]=>[3,2,5,7,1,4,6]=>[1,5,2,3,6,7,4]=>[5,1,6,2,3,7,4]=>000000
[[1,3,6],[2,5,7],[4]]=>[4,2,5,7,1,3,6]=>[5,2,1,3,6,7,4]=>[5,2,6,1,3,7,4]=>000000
[[1,2,6],[3,5,7],[4]]=>[4,3,5,7,1,2,6]=>[2,1,5,3,6,7,4]=>[5,6,2,1,3,7,4]=>000000
[[1,3,6],[2,4,7],[5]]=>[5,2,4,7,1,3,6]=>[5,2,3,1,6,7,4]=>[5,2,6,3,1,7,4]=>000000
[[1,2,6],[3,4,7],[5]]=>[5,3,4,7,1,2,6]=>[2,5,3,1,6,7,4]=>[5,6,2,3,1,7,4]=>000000
[[1,4,5],[2,6,7],[3]]=>[3,2,6,7,1,4,5]=>[1,5,6,2,3,7,4]=>[5,1,6,2,7,3,4]=>000000
[[1,3,5],[2,6,7],[4]]=>[4,2,6,7,1,3,5]=>[5,1,6,2,3,7,4]=>[5,1,6,7,2,3,4]=>000000
[[1,2,5],[3,6,7],[4]]=>[4,3,6,7,1,2,5]=>[2,1,5,6,3,7,4]=>[5,6,2,1,7,3,4]=>000000
[[1,3,4],[2,6,7],[5]]=>[5,2,6,7,1,3,4]=>[5,6,2,1,3,7,4]=>[5,2,6,7,1,3,4]=>000000
[[1,2,4],[3,6,7],[5]]=>[5,3,6,7,1,2,4]=>[2,5,1,6,3,7,4]=>[5,6,2,7,1,3,4]=>000000
[[1,2,3],[4,6,7],[5]]=>[5,4,6,7,1,2,3]=>[5,2,1,6,3,7,4]=>[5,6,7,2,1,3,4]=>000000
[[1,3,5],[2,4,7],[6]]=>[6,2,4,7,1,3,5]=>[5,2,3,6,1,7,4]=>[5,2,6,3,7,1,4]=>000000
[[1,2,5],[3,4,7],[6]]=>[6,3,4,7,1,2,5]=>[2,5,3,6,1,7,4]=>[5,6,2,3,7,1,4]=>000000
[[1,3,4],[2,5,7],[6]]=>[6,2,5,7,1,3,4]=>[5,6,2,3,1,7,4]=>[5,2,6,7,3,1,4]=>000000
[[1,2,4],[3,5,7],[6]]=>[6,3,5,7,1,2,4]=>[2,5,6,3,1,7,4]=>[5,6,2,7,3,1,4]=>000000
[[1,2,3],[4,5,7],[6]]=>[6,4,5,7,1,2,3]=>[5,2,6,3,1,7,4]=>[5,6,7,2,3,1,4]=>000000
[[1,3,5],[2,4,6],[7]]=>[7,2,4,6,1,3,5]=>[5,2,3,6,7,4,1]=>[5,2,6,3,7,4,1]=>000000
[[1,2,5],[3,4,6],[7]]=>[7,3,4,6,1,2,5]=>[2,5,3,6,7,4,1]=>[5,6,2,3,7,4,1]=>000000
[[1,3,4],[2,5,6],[7]]=>[7,2,5,6,1,3,4]=>[5,6,2,3,7,4,1]=>[5,2,6,7,3,4,1]=>000000
[[1,2,4],[3,5,6],[7]]=>[7,3,5,6,1,2,4]=>[2,5,6,3,7,4,1]=>[5,6,2,7,3,4,1]=>000000
[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[5,2,6,3,7,4,1]=>[5,6,7,2,3,4,1]=>000000
[[1,4,7],[2,5],[3,6]]=>[3,6,2,5,1,4,7]=>[1,3,5,6,4,2,7]=>[5,3,1,6,4,2,7]=>000001
[[1,3,7],[2,5],[4,6]]=>[4,6,2,5,1,3,7]=>[3,1,5,6,4,2,7]=>[5,3,6,1,4,2,7]=>000001
[[1,2,7],[3,5],[4,6]]=>[4,6,3,5,1,2,7]=>[1,5,3,6,4,2,7]=>[5,6,3,1,4,2,7]=>000001
[[1,3,7],[2,4],[5,6]]=>[5,6,2,4,1,3,7]=>[3,5,1,6,4,2,7]=>[5,3,6,4,1,2,7]=>000001
[[1,2,7],[3,4],[5,6]]=>[5,6,3,4,1,2,7]=>[5,3,1,6,4,2,7]=>[5,6,3,4,1,2,7]=>000001
[[1,4,6],[2,5],[3,7]]=>[3,7,2,5,1,4,6]=>[1,3,5,6,4,7,2]=>[5,3,1,6,4,7,2]=>000000
[[1,3,6],[2,5],[4,7]]=>[4,7,2,5,1,3,6]=>[3,1,5,6,4,7,2]=>[5,3,6,1,4,7,2]=>000000
[[1,2,6],[3,5],[4,7]]=>[4,7,3,5,1,2,6]=>[1,5,3,6,4,7,2]=>[5,6,3,1,4,7,2]=>000000
[[1,3,6],[2,4],[5,7]]=>[5,7,2,4,1,3,6]=>[3,5,1,6,4,7,2]=>[5,3,6,4,1,7,2]=>000000
[[1,2,6],[3,4],[5,7]]=>[5,7,3,4,1,2,6]=>[5,3,1,6,4,7,2]=>[5,6,3,4,1,7,2]=>000000
[[1,4,5],[2,6],[3,7]]=>[3,7,2,6,1,4,5]=>[1,3,5,6,7,4,2]=>[5,3,1,6,7,4,2]=>000000
[[1,3,5],[2,6],[4,7]]=>[4,7,2,6,1,3,5]=>[3,1,5,6,7,4,2]=>[5,3,6,1,7,4,2]=>000000
[[1,2,5],[3,6],[4,7]]=>[4,7,3,6,1,2,5]=>[1,5,3,6,7,4,2]=>[5,6,3,1,7,4,2]=>000000
[[1,3,4],[2,6],[5,7]]=>[5,7,2,6,1,3,4]=>[3,5,1,6,7,4,2]=>[5,3,6,7,1,4,2]=>000000
[[1,2,4],[3,6],[5,7]]=>[5,7,3,6,1,2,4]=>[5,3,1,6,7,4,2]=>[5,6,3,7,1,4,2]=>000000
[[1,2,3],[4,6],[5,7]]=>[5,7,4,6,1,2,3]=>[5,1,6,3,7,4,2]=>[5,6,7,3,1,4,2]=>000000
[[1,3,5],[2,4],[6,7]]=>[6,7,2,4,1,3,5]=>[3,5,6,4,1,7,2]=>[5,3,6,4,7,1,2]=>000000
[[1,2,5],[3,4],[6,7]]=>[6,7,3,4,1,2,5]=>[5,3,6,4,1,7,2]=>[5,6,3,4,7,1,2]=>000000
[[1,3,4],[2,5],[6,7]]=>[6,7,2,5,1,3,4]=>[3,5,6,1,7,4,2]=>[5,3,6,7,4,1,2]=>000000
[[1,2,4],[3,5],[6,7]]=>[6,7,3,5,1,2,4]=>[5,3,6,1,7,4,2]=>[5,6,3,7,4,1,2]=>000000
[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[5,6,3,1,7,4,2]=>[5,6,7,3,4,1,2]=>000000
[[1,5,7],[2,6],[3],[4]]=>[4,3,2,6,1,5,7]=>[3,2,1,5,6,4,7]=>[5,3,2,1,6,4,7]=>000001
[[1,4,7],[2,6],[3],[5]]=>[5,3,2,6,1,4,7]=>[3,2,5,1,6,4,7]=>[5,3,2,6,1,4,7]=>000001
[[1,3,7],[2,6],[4],[5]]=>[5,4,2,6,1,3,7]=>[3,5,2,1,6,4,7]=>[5,3,6,2,1,4,7]=>000001
[[1,2,7],[3,6],[4],[5]]=>[5,4,3,6,1,2,7]=>[5,3,2,1,6,4,7]=>[5,6,3,2,1,4,7]=>000001
[[1,4,7],[2,5],[3],[6]]=>[6,3,2,5,1,4,7]=>[3,2,5,6,4,1,7]=>[5,3,2,6,4,1,7]=>000001
[[1,3,7],[2,5],[4],[6]]=>[6,4,2,5,1,3,7]=>[3,5,2,6,4,1,7]=>[5,3,6,2,4,1,7]=>000001
[[1,2,7],[3,5],[4],[6]]=>[6,4,3,5,1,2,7]=>[5,3,2,6,4,1,7]=>[5,6,3,2,4,1,7]=>000001
[[1,3,7],[2,4],[5],[6]]=>[6,5,2,4,1,3,7]=>[3,5,6,4,2,1,7]=>[5,3,6,4,2,1,7]=>000001
[[1,2,7],[3,4],[5],[6]]=>[6,5,3,4,1,2,7]=>[5,3,6,4,2,1,7]=>[5,6,3,4,2,1,7]=>000001
[[1,5,6],[2,7],[3],[4]]=>[4,3,2,7,1,5,6]=>[3,2,1,5,6,7,4]=>[5,3,2,1,6,7,4]=>000000
[[1,4,6],[2,7],[3],[5]]=>[5,3,2,7,1,4,6]=>[3,2,5,1,6,7,4]=>[5,3,2,6,1,7,4]=>000000
[[1,3,6],[2,7],[4],[5]]=>[5,4,2,7,1,3,6]=>[3,5,2,1,6,7,4]=>[5,3,6,2,1,7,4]=>000000
[[1,2,6],[3,7],[4],[5]]=>[5,4,3,7,1,2,6]=>[5,3,2,1,6,7,4]=>[5,6,3,2,1,7,4]=>000000
[[1,4,5],[2,7],[3],[6]]=>[6,3,2,7,1,4,5]=>[3,2,5,6,1,7,4]=>[5,3,2,6,7,1,4]=>000000
[[1,3,5],[2,7],[4],[6]]=>[6,4,2,7,1,3,5]=>[3,5,2,6,1,7,4]=>[5,3,6,2,7,1,4]=>000000
[[1,2,5],[3,7],[4],[6]]=>[6,4,3,7,1,2,5]=>[5,3,2,6,1,7,4]=>[5,6,3,2,7,1,4]=>000000
[[1,3,4],[2,7],[5],[6]]=>[6,5,2,7,1,3,4]=>[3,5,6,2,1,7,4]=>[5,3,6,7,2,1,4]=>000000
[[1,2,4],[3,7],[5],[6]]=>[6,5,3,7,1,2,4]=>[5,3,6,2,1,7,4]=>[5,6,3,7,2,1,4]=>000000
[[1,2,3],[4,7],[5],[6]]=>[6,5,4,7,1,2,3]=>[5,6,3,2,1,7,4]=>[5,6,7,3,2,1,4]=>000000
[[1,4,6],[2,5],[3],[7]]=>[7,3,2,5,1,4,6]=>[3,2,5,6,4,7,1]=>[5,3,2,6,4,7,1]=>000000
[[1,3,6],[2,5],[4],[7]]=>[7,4,2,5,1,3,6]=>[3,5,2,6,4,7,1]=>[5,3,6,2,4,7,1]=>000000
[[1,2,6],[3,5],[4],[7]]=>[7,4,3,5,1,2,6]=>[5,3,2,6,4,7,1]=>[5,6,3,2,4,7,1]=>000000
[[1,3,6],[2,4],[5],[7]]=>[7,5,2,4,1,3,6]=>[3,5,6,4,2,7,1]=>[5,3,6,4,2,7,1]=>000000
[[1,2,6],[3,4],[5],[7]]=>[7,5,3,4,1,2,6]=>[5,3,6,4,2,7,1]=>[5,6,3,4,2,7,1]=>000000
[[1,4,5],[2,6],[3],[7]]=>[7,3,2,6,1,4,5]=>[3,2,5,6,7,4,1]=>[5,3,2,6,7,4,1]=>000000
[[1,3,5],[2,6],[4],[7]]=>[7,4,2,6,1,3,5]=>[3,5,2,6,7,4,1]=>[5,3,6,2,7,4,1]=>000000
[[1,2,5],[3,6],[4],[7]]=>[7,4,3,6,1,2,5]=>[5,3,2,6,7,4,1]=>[5,6,3,2,7,4,1]=>000000
[[1,3,4],[2,6],[5],[7]]=>[7,5,2,6,1,3,4]=>[3,5,6,2,7,4,1]=>[5,3,6,7,2,4,1]=>000000
[[1,2,4],[3,6],[5],[7]]=>[7,5,3,6,1,2,4]=>[5,3,6,2,7,4,1]=>[5,6,3,7,2,4,1]=>000000
[[1,2,3],[4,6],[5],[7]]=>[7,5,4,6,1,2,3]=>[5,6,3,2,7,4,1]=>[5,6,7,3,2,4,1]=>000000
[[1,3,5],[2,4],[6],[7]]=>[7,6,2,4,1,3,5]=>[3,5,6,4,7,2,1]=>[5,3,6,4,7,2,1]=>000000
[[1,2,5],[3,4],[6],[7]]=>[7,6,3,4,1,2,5]=>[5,3,6,4,7,2,1]=>[5,6,3,4,7,2,1]=>000000
[[1,3,4],[2,5],[6],[7]]=>[7,6,2,5,1,3,4]=>[3,5,6,7,4,2,1]=>[5,3,6,7,4,2,1]=>000000
[[1,2,4],[3,5],[6],[7]]=>[7,6,3,5,1,2,4]=>[5,3,6,7,4,2,1]=>[5,6,3,7,4,2,1]=>000000
[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[5,6,3,7,4,2,1]=>[5,6,7,3,4,2,1]=>000000
[[1,6,7],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7]=>[5,4,3,2,1,6,7]=>[5,4,3,2,1,6,7]=>000011
[[1,5,7],[2],[3],[4],[6]]=>[6,4,3,2,1,5,7]=>[5,4,3,2,6,1,7]=>[5,4,3,2,6,1,7]=>000001
[[1,4,7],[2],[3],[5],[6]]=>[6,5,3,2,1,4,7]=>[5,4,3,6,2,1,7]=>[5,4,3,6,2,1,7]=>000001
[[1,3,7],[2],[4],[5],[6]]=>[6,5,4,2,1,3,7]=>[5,4,6,3,2,1,7]=>[5,4,6,3,2,1,7]=>000001
[[1,2,7],[3],[4],[5],[6]]=>[6,5,4,3,1,2,7]=>[5,6,4,3,2,1,7]=>[5,6,4,3,2,1,7]=>000001
[[1,5,6],[2],[3],[4],[7]]=>[7,4,3,2,1,5,6]=>[5,4,3,2,6,7,1]=>[5,4,3,2,6,7,1]=>000000
[[1,4,6],[2],[3],[5],[7]]=>[7,5,3,2,1,4,6]=>[5,4,3,6,2,7,1]=>[5,4,3,6,2,7,1]=>000000
[[1,3,6],[2],[4],[5],[7]]=>[7,5,4,2,1,3,6]=>[5,4,6,3,2,7,1]=>[5,4,6,3,2,7,1]=>000000
[[1,2,6],[3],[4],[5],[7]]=>[7,5,4,3,1,2,6]=>[5,6,4,3,2,7,1]=>[5,6,4,3,2,7,1]=>000000
[[1,4,5],[2],[3],[6],[7]]=>[7,6,3,2,1,4,5]=>[5,4,3,6,7,2,1]=>[5,4,3,6,7,2,1]=>000000
[[1,3,5],[2],[4],[6],[7]]=>[7,6,4,2,1,3,5]=>[5,4,6,3,7,2,1]=>[5,4,6,3,7,2,1]=>000000
[[1,2,5],[3],[4],[6],[7]]=>[7,6,4,3,1,2,5]=>[5,6,4,3,7,2,1]=>[5,6,4,3,7,2,1]=>000000
[[1,3,4],[2],[5],[6],[7]]=>[7,6,5,2,1,3,4]=>[5,4,6,7,3,2,1]=>[5,4,6,7,3,2,1]=>000000
[[1,2,4],[3],[5],[6],[7]]=>[7,6,5,3,1,2,4]=>[5,6,4,7,3,2,1]=>[5,6,4,7,3,2,1]=>000000
[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[5,6,7,4,3,2,1]=>[5,6,7,4,3,2,1]=>000000
[[1,5],[2,6],[3,7],[4]]=>[4,3,7,2,6,1,5]=>[2,1,4,6,7,5,3]=>[6,4,2,1,7,5,3]=>000000
[[1,4],[2,6],[3,7],[5]]=>[5,3,7,2,6,1,4]=>[2,4,1,6,7,5,3]=>[6,4,2,7,1,5,3]=>000000
[[1,3],[2,6],[4,7],[5]]=>[5,4,7,2,6,1,3]=>[4,2,1,6,7,5,3]=>[6,4,7,2,1,5,3]=>000000
[[1,2],[3,6],[4,7],[5]]=>[5,4,7,3,6,1,2]=>[2,1,6,4,7,5,3]=>[6,7,4,2,1,5,3]=>000000
[[1,4],[2,5],[3,7],[6]]=>[6,3,7,2,5,1,4]=>[2,4,6,1,7,5,3]=>[6,4,2,7,5,1,3]=>000000
[[1,3],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3]=>[4,2,6,1,7,5,3]=>[6,4,7,2,5,1,3]=>000000
[[1,2],[3,5],[4,7],[6]]=>[6,4,7,3,5,1,2]=>[2,6,4,1,7,5,3]=>[6,7,4,2,5,1,3]=>000000
[[1,3],[2,4],[5,7],[6]]=>[6,5,7,2,4,1,3]=>[4,6,2,1,7,5,3]=>[6,4,7,5,2,1,3]=>000000
[[1,2],[3,4],[5,7],[6]]=>[6,5,7,3,4,1,2]=>[6,4,2,1,7,5,3]=>[6,7,4,5,2,1,3]=>000000
[[1,4],[2,5],[3,6],[7]]=>[7,3,6,2,5,1,4]=>[2,4,6,7,5,3,1]=>[6,4,2,7,5,3,1]=>000000
[[1,3],[2,5],[4,6],[7]]=>[7,4,6,2,5,1,3]=>[4,2,6,7,5,3,1]=>[6,4,7,2,5,3,1]=>000000
[[1,2],[3,5],[4,6],[7]]=>[7,4,6,3,5,1,2]=>[2,6,4,7,5,3,1]=>[6,7,4,2,5,3,1]=>000000
[[1,3],[2,4],[5,6],[7]]=>[7,5,6,2,4,1,3]=>[4,6,2,7,5,3,1]=>[6,4,7,5,2,3,1]=>000000
[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[6,4,2,7,5,3,1]=>[6,7,4,5,2,3,1]=>000000
[[1,6],[2,7],[3],[4],[5]]=>[5,4,3,2,7,1,6]=>[4,3,2,1,6,7,5]=>[6,4,3,2,1,7,5]=>000000
[[1,5],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5]=>[4,3,2,6,1,7,5]=>[6,4,3,2,7,1,5]=>000000
[[1,4],[2,7],[3],[5],[6]]=>[6,5,3,2,7,1,4]=>[4,3,6,2,1,7,5]=>[6,4,3,7,2,1,5]=>000000
[[1,3],[2,7],[4],[5],[6]]=>[6,5,4,2,7,1,3]=>[4,6,3,2,1,7,5]=>[6,4,7,3,2,1,5]=>000000
[[1,2],[3,7],[4],[5],[6]]=>[6,5,4,3,7,1,2]=>[6,4,3,2,1,7,5]=>[6,7,4,3,2,1,5]=>000000
[[1,5],[2,6],[3],[4],[7]]=>[7,4,3,2,6,1,5]=>[4,3,2,6,7,5,1]=>[6,4,3,2,7,5,1]=>000000
[[1,4],[2,6],[3],[5],[7]]=>[7,5,3,2,6,1,4]=>[4,3,6,2,7,5,1]=>[6,4,3,7,2,5,1]=>000000
[[1,3],[2,6],[4],[5],[7]]=>[7,5,4,2,6,1,3]=>[4,6,3,2,7,5,1]=>[6,4,7,3,2,5,1]=>000000
[[1,2],[3,6],[4],[5],[7]]=>[7,5,4,3,6,1,2]=>[6,4,3,2,7,5,1]=>[6,7,4,3,2,5,1]=>000000
[[1,4],[2,5],[3],[6],[7]]=>[7,6,3,2,5,1,4]=>[4,3,6,7,5,2,1]=>[6,4,3,7,5,2,1]=>000000
[[1,3],[2,5],[4],[6],[7]]=>[7,6,4,2,5,1,3]=>[4,6,3,7,5,2,1]=>[6,4,7,3,5,2,1]=>000000
[[1,2],[3,5],[4],[6],[7]]=>[7,6,4,3,5,1,2]=>[6,4,3,7,5,2,1]=>[6,7,4,3,5,2,1]=>000000
[[1,3],[2,4],[5],[6],[7]]=>[7,6,5,2,4,1,3]=>[4,6,7,5,3,2,1]=>[6,4,7,5,3,2,1]=>000000
[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[6,4,7,5,3,2,1]=>[6,7,4,5,3,2,1]=>000000
[[1,7],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7]=>[6,5,4,3,2,1,7]=>[6,5,4,3,2,1,7]=>000001
[[1,6],[2],[3],[4],[5],[7]]=>[7,5,4,3,2,1,6]=>[6,5,4,3,2,7,1]=>[6,5,4,3,2,7,1]=>000000
[[1,5],[2],[3],[4],[6],[7]]=>[7,6,4,3,2,1,5]=>[6,5,4,3,7,2,1]=>[6,5,4,3,7,2,1]=>000000
[[1,4],[2],[3],[5],[6],[7]]=>[7,6,5,3,2,1,4]=>[6,5,4,7,3,2,1]=>[6,5,4,7,3,2,1]=>000000
[[1,3],[2],[4],[5],[6],[7]]=>[7,6,5,4,2,1,3]=>[6,5,7,4,3,2,1]=>[6,5,7,4,3,2,1]=>000000
[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[6,7,5,4,3,2,1]=>[6,7,5,4,3,2,1]=>000000
[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[7,6,5,4,3,2,1]=>[7,6,5,4,3,2,1]=>000000
[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8]=>1111111
[[1,3,4,5,6,7,8],[2]]=>[2,1,3,4,5,6,7,8]=>[2,1,3,4,5,6,7,8]=>[2,1,3,4,5,6,7,8]=>0111111
[[1,2,4,5,6,7,8],[3]]=>[3,1,2,4,5,6,7,8]=>[2,3,1,4,5,6,7,8]=>[2,3,1,4,5,6,7,8]=>0011111
[[1,2,3,5,6,7,8],[4]]=>[4,1,2,3,5,6,7,8]=>[2,3,4,1,5,6,7,8]=>[2,3,4,1,5,6,7,8]=>0001111
[[1,2,3,4,6,7,8],[5]]=>[5,1,2,3,4,6,7,8]=>[2,3,4,5,1,6,7,8]=>[2,3,4,5,1,6,7,8]=>0000111
[[1,2,3,4,5,6,7],[8]]=>[8,1,2,3,4,5,6,7]=>[2,3,4,5,6,7,8,1]=>[2,3,4,5,6,7,8,1]=>0000000
[[1,3,5,6,7,8],[2,4]]=>[2,4,1,3,5,6,7,8]=>[1,3,4,2,5,6,7,8]=>[3,1,4,2,5,6,7,8]=>0001111
[[1,2,5,6,7,8],[3,4]]=>[3,4,1,2,5,6,7,8]=>[3,1,4,2,5,6,7,8]=>[3,4,1,2,5,6,7,8]=>0001111
[[1,2,3,4,6,8],[5,7]]=>[5,7,1,2,3,4,6,8]=>[3,4,5,1,6,7,2,8]=>[3,4,5,6,1,7,2,8]=>0000001
[[1,2,3,4,5,8],[6,7]]=>[6,7,1,2,3,4,5,8]=>[3,4,5,6,1,7,2,8]=>[3,4,5,6,7,1,2,8]=>0000001
[[1,2,3,5,6,7],[4,8]]=>[4,8,1,2,3,5,6,7]=>[3,4,1,5,6,7,8,2]=>[3,4,5,1,6,7,8,2]=>0000000
[[1,2,3,4,6,7],[5,8]]=>[5,8,1,2,3,4,6,7]=>[3,4,5,1,6,7,8,2]=>[3,4,5,6,1,7,8,2]=>0000000
[[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>[3,4,5,6,7,1,8,2]=>[3,4,5,6,7,8,1,2]=>0000000
[[1,4,5,6,7,8],[2],[3]]=>[3,2,1,4,5,6,7,8]=>[3,2,1,4,5,6,7,8]=>[3,2,1,4,5,6,7,8]=>0011111
[[1,3,5,6,7,8],[2],[4]]=>[4,2,1,3,5,6,7,8]=>[3,2,4,1,5,6,7,8]=>[3,2,4,1,5,6,7,8]=>0001111
[[1,2,5,6,7,8],[3],[4]]=>[4,3,1,2,5,6,7,8]=>[3,4,2,1,5,6,7,8]=>[3,4,2,1,5,6,7,8]=>0001111
[[1,3,4,6,7,8],[2],[5]]=>[5,2,1,3,4,6,7,8]=>[3,2,4,5,1,6,7,8]=>[3,2,4,5,1,6,7,8]=>0000111
[[1,2,4,6,7,8],[3],[5]]=>[5,3,1,2,4,6,7,8]=>[3,4,2,5,1,6,7,8]=>[3,4,2,5,1,6,7,8]=>0000111
[[1,2,3,6,7,8],[4],[5]]=>[5,4,1,2,3,6,7,8]=>[3,4,5,2,1,6,7,8]=>[3,4,5,2,1,6,7,8]=>0000111
[[1,2,3,4,6,8],[5],[7]]=>[7,5,1,2,3,4,6,8]=>[3,4,5,6,2,7,1,8]=>[3,4,5,6,2,7,1,8]=>0000001
[[1,2,4,5,8],[3,6,7]]=>[3,6,7,1,2,4,5,8]=>[1,4,5,6,2,7,3,8]=>[4,5,1,6,7,2,3,8]=>0000001
[[1,2,5,6,7],[3,4,8]]=>[3,4,8,1,2,5,6,7]=>[1,4,2,5,6,7,8,3]=>[4,5,1,2,6,7,8,3]=>0000000
[[1,3,6,7,8],[2,4],[5]]=>[5,2,4,1,3,6,7,8]=>[2,4,5,3,1,6,7,8]=>[4,2,5,3,1,6,7,8]=>0000111
[[1,2,6,7,8],[3,4],[5]]=>[5,3,4,1,2,6,7,8]=>[4,2,5,3,1,6,7,8]=>[4,5,2,3,1,6,7,8]=>0000111
[[1,2,3,6,8],[4,7],[5]]=>[5,4,7,1,2,3,6,8]=>[4,5,2,1,6,7,3,8]=>[4,5,6,2,1,7,3,8]=>0000001
[[1,3,4,5,8],[2,7],[6]]=>[6,2,7,1,3,4,5,8]=>[2,4,5,6,1,7,3,8]=>[4,2,5,6,7,1,3,8]=>0000001
[[1,2,4,5,8],[3,7],[6]]=>[6,3,7,1,2,4,5,8]=>[4,2,5,6,1,7,3,8]=>[4,5,2,6,7,1,3,8]=>0000001
[[1,2,3,5,8],[4,6],[7]]=>[7,4,6,1,2,3,5,8]=>[4,5,2,6,7,3,1,8]=>[4,5,6,2,7,3,1,8]=>0000001
[[1,2,3,6,7],[4,8],[5]]=>[5,4,8,1,2,3,6,7]=>[4,5,2,1,6,7,8,3]=>[4,5,6,2,1,7,8,3]=>0000000
[[1,2,3,4,7],[5,8],[6]]=>[6,5,8,1,2,3,4,7]=>[4,5,6,2,1,7,8,3]=>[4,5,6,7,2,1,8,3]=>0000000
[[1,2,3,4,5],[6,8],[7]]=>[7,6,8,1,2,3,4,5]=>[4,5,6,7,2,1,8,3]=>[4,5,6,7,8,2,1,3]=>0000000
[[1,2,3,4,6],[5,7],[8]]=>[8,5,7,1,2,3,4,6]=>[4,5,6,2,7,8,3,1]=>[4,5,6,7,2,8,3,1]=>0000000
[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[4,5,6,7,2,8,3,1]=>[4,5,6,7,8,2,3,1]=>0000000
[[1,5,6,7,8],[2],[3],[4]]=>[4,3,2,1,5,6,7,8]=>[4,3,2,1,5,6,7,8]=>[4,3,2,1,5,6,7,8]=>0001111
[[1,4,6,7,8],[2],[3],[5]]=>[5,3,2,1,4,6,7,8]=>[4,3,2,5,1,6,7,8]=>[4,3,2,5,1,6,7,8]=>0000111
[[1,3,6,7,8],[2],[4],[5]]=>[5,4,2,1,3,6,7,8]=>[4,3,5,2,1,6,7,8]=>[4,3,5,2,1,6,7,8]=>0000111
[[1,2,6,7,8],[3],[4],[5]]=>[5,4,3,1,2,6,7,8]=>[4,5,3,2,1,6,7,8]=>[4,5,3,2,1,6,7,8]=>0000111
[[1,2,5,7,8],[3],[4],[6]]=>[6,4,3,1,2,5,7,8]=>[4,5,3,2,6,1,7,8]=>[4,5,3,2,6,1,7,8]=>0000011
[[1,2,4,6,8],[3],[5],[7]]=>[7,5,3,1,2,4,6,8]=>[4,5,3,6,2,7,1,8]=>[4,5,3,6,2,7,1,8]=>0000001
[[1,2,3,4,8],[5],[6],[7]]=>[7,6,5,1,2,3,4,8]=>[4,5,6,7,3,2,1,8]=>[4,5,6,7,3,2,1,8]=>0000001
[[1,2,3,4,7],[5],[6],[8]]=>[8,6,5,1,2,3,4,7]=>[4,5,6,7,3,2,8,1]=>[4,5,6,7,3,2,8,1]=>0000000
[[1,2,3,4,6],[5],[7],[8]]=>[8,7,5,1,2,3,4,6]=>[4,5,6,7,3,8,2,1]=>[4,5,6,7,3,8,2,1]=>0000000
[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[4,5,6,7,8,3,2,1]=>[4,5,6,7,8,3,2,1]=>0000000
[[1,2,3,7],[4,5,6,8]]=>[4,5,6,8,1,2,3,7]=>[1,5,2,6,3,7,8,4]=>[5,6,7,1,2,3,8,4]=>0000000
[[1,2,4,5],[3,6,7,8]]=>[3,6,7,8,1,2,4,5]=>[5,6,1,2,7,3,8,4]=>[5,6,1,7,8,2,3,4]=>0000000
[[1,2,3,5],[4,6,7,8]]=>[4,6,7,8,1,2,3,5]=>[1,5,6,2,7,3,8,4]=>[5,6,7,1,8,2,3,4]=>0000000
[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>[5,6,7,8,1,2,3,4]=>0000000
[[1,3,6,8],[2,5,7],[4]]=>[4,2,5,7,1,3,6,8]=>[5,2,1,3,6,7,4,8]=>[5,2,6,1,3,7,4,8]=>0000001
[[1,2,6,8],[3,5,7],[4]]=>[4,3,5,7,1,2,6,8]=>[2,1,5,3,6,7,4,8]=>[5,6,2,1,3,7,4,8]=>0000001
[[1,3,4,8],[2,6,7],[5]]=>[5,2,6,7,1,3,4,8]=>[5,6,2,1,3,7,4,8]=>[5,2,6,7,1,3,4,8]=>0000001
[[1,2,4,8],[3,6,7],[5]]=>[5,3,6,7,1,2,4,8]=>[2,5,1,6,3,7,4,8]=>[5,6,2,7,1,3,4,8]=>0000001
[[1,2,3,8],[4,6,7],[5]]=>[5,4,6,7,1,2,3,8]=>[5,2,1,6,3,7,4,8]=>[5,6,7,2,1,3,4,8]=>0000001
[[1,2,3,8],[4,5,6],[7]]=>[7,4,5,6,1,2,3,8]=>[5,2,6,3,7,4,1,8]=>[5,6,7,2,3,4,1,8]=>0000001
[[1,3,6,7],[2,4,8],[5]]=>[5,2,4,8,1,3,6,7]=>[5,2,3,1,6,7,8,4]=>[5,2,6,3,1,7,8,4]=>0000000
[[1,4,5,7],[2,6,8],[3]]=>[3,2,6,8,1,4,5,7]=>[1,5,6,2,3,7,8,4]=>[5,1,6,2,7,3,8,4]=>0000000
[[1,2,5,7],[3,6,8],[4]]=>[4,3,6,8,1,2,5,7]=>[2,1,5,6,3,7,8,4]=>[5,6,2,1,7,3,8,4]=>0000000
[[1,3,5,7],[2,4,8],[6]]=>[6,2,4,8,1,3,5,7]=>[5,2,3,6,1,7,8,4]=>[5,2,6,3,7,1,8,4]=>0000000
[[1,2,4,7],[3,5,8],[6]]=>[6,3,5,8,1,2,4,7]=>[2,5,6,3,1,7,8,4]=>[5,6,2,7,3,1,8,4]=>0000000
[[1,2,5,6],[3,7,8],[4]]=>[4,3,7,8,1,2,5,6]=>[2,1,5,6,7,3,8,4]=>[5,6,2,1,7,8,3,4]=>0000000
[[1,2,4,5],[3,7,8],[6]]=>[6,3,7,8,1,2,4,5]=>[2,5,6,1,7,3,8,4]=>[5,6,2,7,8,1,3,4]=>0000000
[[1,2,3,5],[4,7,8],[6]]=>[6,4,7,8,1,2,3,5]=>[5,2,6,1,7,3,8,4]=>[5,6,7,2,8,1,3,4]=>0000000
[[1,2,3,4],[5,7,8],[6]]=>[6,5,7,8,1,2,3,4]=>[5,6,2,1,7,3,8,4]=>[5,6,7,8,2,1,3,4]=>0000000
[[1,2,4,6],[3,5,8],[7]]=>[7,3,5,8,1,2,4,6]=>[2,5,6,3,7,1,8,4]=>[5,6,2,7,3,8,1,4]=>0000000
[[1,2,3,6],[4,5,8],[7]]=>[7,4,5,8,1,2,3,6]=>[5,2,6,3,7,1,8,4]=>[5,6,7,2,3,8,1,4]=>0000000
[[1,2,3,4],[5,6,8],[7]]=>[7,5,6,8,1,2,3,4]=>[5,6,2,7,3,1,8,4]=>[5,6,7,8,2,3,1,4]=>0000000
[[1,2,3,7],[4,5,6],[8]]=>[8,4,5,6,1,2,3,7]=>[5,2,6,3,7,4,8,1]=>[5,6,7,2,3,4,8,1]=>0000000
[[1,3,5,6],[2,4,7],[8]]=>[8,2,4,7,1,3,5,6]=>[5,2,3,6,7,8,4,1]=>[5,2,6,3,7,8,4,1]=>0000000
[[1,2,4,6],[3,5,7],[8]]=>[8,3,5,7,1,2,4,6]=>[2,5,6,3,7,8,4,1]=>[5,6,2,7,3,8,4,1]=>0000000
[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[5,6,2,7,3,8,4,1]=>[5,6,7,8,2,3,4,1]=>0000000
[[1,3,7,8],[2,5],[4,6]]=>[4,6,2,5,1,3,7,8]=>[3,1,5,6,4,2,7,8]=>[5,3,6,1,4,2,7,8]=>0000011
[[1,2,7,8],[3,5],[4,6]]=>[4,6,3,5,1,2,7,8]=>[1,5,3,6,4,2,7,8]=>[5,6,3,1,4,2,7,8]=>0000011
[[1,3,7,8],[2,4],[5,6]]=>[5,6,2,4,1,3,7,8]=>[3,5,1,6,4,2,7,8]=>[5,3,6,4,1,2,7,8]=>0000011
[[1,2,7,8],[3,4],[5,6]]=>[5,6,3,4,1,2,7,8]=>[5,3,1,6,4,2,7,8]=>[5,6,3,4,1,2,7,8]=>0000011
[[1,3,6,8],[2,4],[5,7]]=>[5,7,2,4,1,3,6,8]=>[3,5,1,6,4,7,2,8]=>[5,3,6,4,1,7,2,8]=>0000001
[[1,2,6,8],[3,4],[5,7]]=>[5,7,3,4,1,2,6,8]=>[5,3,1,6,4,7,2,8]=>[5,6,3,4,1,7,2,8]=>0000001
[[1,2,5,7],[3,4],[6,8]]=>[6,8,3,4,1,2,5,7]=>[5,3,6,4,1,7,8,2]=>[5,6,3,4,7,1,8,2]=>0000000
[[1,2,4,7],[3,5],[6,8]]=>[6,8,3,5,1,2,4,7]=>[5,3,6,1,7,4,8,2]=>[5,6,3,7,4,1,8,2]=>0000000
[[1,4,5,6],[2,7],[3,8]]=>[3,8,2,7,1,4,5,6]=>[1,3,5,6,7,8,4,2]=>[5,3,1,6,7,8,4,2]=>0000000
[[1,3,5,6],[2,7],[4,8]]=>[4,8,2,7,1,3,5,6]=>[3,1,5,6,7,8,4,2]=>[5,3,6,1,7,8,4,2]=>0000000
[[1,2,3,6],[4,7],[5,8]]=>[5,8,4,7,1,2,3,6]=>[5,1,6,3,7,8,4,2]=>[5,6,7,3,1,8,4,2]=>0000000
[[1,3,4,5],[2,7],[6,8]]=>[6,8,2,7,1,3,4,5]=>[3,5,6,1,7,8,4,2]=>[5,3,6,7,8,1,4,2]=>0000000
[[1,2,3,4],[5,7],[6,8]]=>[6,8,5,7,1,2,3,4]=>[5,6,1,7,3,8,4,2]=>[5,6,7,8,3,1,4,2]=>0000000
[[1,3,5,6],[2,4],[7,8]]=>[7,8,2,4,1,3,5,6]=>[3,5,6,4,7,1,8,2]=>[5,3,6,4,7,8,1,2]=>0000000
[[1,2,5,6],[3,4],[7,8]]=>[7,8,3,4,1,2,5,6]=>[5,3,6,4,7,1,8,2]=>[5,6,3,4,7,8,1,2]=>0000000
[[1,2,4,6],[3,5],[7,8]]=>[7,8,3,5,1,2,4,6]=>[5,3,6,7,4,1,8,2]=>[5,6,3,7,4,8,1,2]=>0000000
[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[5,6,7,3,1,8,4,2]=>[5,6,7,8,3,4,1,2]=>0000000
[[1,5,7,8],[2,6],[3],[4]]=>[4,3,2,6,1,5,7,8]=>[3,2,1,5,6,4,7,8]=>[5,3,2,1,6,4,7,8]=>0000011
[[1,4,7,8],[2,6],[3],[5]]=>[5,3,2,6,1,4,7,8]=>[3,2,5,1,6,4,7,8]=>[5,3,2,6,1,4,7,8]=>0000011
[[1,3,7,8],[2,6],[4],[5]]=>[5,4,2,6,1,3,7,8]=>[3,5,2,1,6,4,7,8]=>[5,3,6,2,1,4,7,8]=>0000011
[[1,2,7,8],[3,6],[4],[5]]=>[5,4,3,6,1,2,7,8]=>[5,3,2,1,6,4,7,8]=>[5,6,3,2,1,4,7,8]=>0000011
[[1,3,7,8],[2,4],[5],[6]]=>[6,5,2,4,1,3,7,8]=>[3,5,6,4,2,1,7,8]=>[5,3,6,4,2,1,7,8]=>0000011
[[1,4,6,8],[2,7],[3],[5]]=>[5,3,2,7,1,4,6,8]=>[3,2,5,1,6,7,4,8]=>[5,3,2,6,1,7,4,8]=>0000001
[[1,3,5,8],[2,7],[4],[6]]=>[6,4,2,7,1,3,5,8]=>[3,5,2,6,1,7,4,8]=>[5,3,6,2,7,1,4,8]=>0000001
[[1,2,5,8],[3,7],[4],[6]]=>[6,4,3,7,1,2,5,8]=>[5,3,2,6,1,7,4,8]=>[5,6,3,2,7,1,4,8]=>0000001
[[1,2,4,8],[3,7],[5],[6]]=>[6,5,3,7,1,2,4,8]=>[5,3,6,2,1,7,4,8]=>[5,6,3,7,2,1,4,8]=>0000001
[[1,3,6,8],[2,5],[4],[7]]=>[7,4,2,5,1,3,6,8]=>[3,5,2,6,4,7,1,8]=>[5,3,6,2,4,7,1,8]=>0000001
[[1,2,6,8],[3,4],[5],[7]]=>[7,5,3,4,1,2,6,8]=>[5,3,6,4,2,7,1,8]=>[5,6,3,4,2,7,1,8]=>0000001
[[1,2,5,8],[3,6],[4],[7]]=>[7,4,3,6,1,2,5,8]=>[5,3,2,6,7,4,1,8]=>[5,6,3,2,7,4,1,8]=>0000001
[[1,2,3,8],[4,5],[6],[7]]=>[7,6,4,5,1,2,3,8]=>[5,6,3,7,4,2,1,8]=>[5,6,7,3,4,2,1,8]=>0000001
[[1,4,6,7],[2,8],[3],[5]]=>[5,3,2,8,1,4,6,7]=>[3,2,5,1,6,7,8,4]=>[5,3,2,6,1,7,8,4]=>0000000
[[1,2,6,7],[3,8],[4],[5]]=>[5,4,3,8,1,2,6,7]=>[5,3,2,1,6,7,8,4]=>[5,6,3,2,1,7,8,4]=>0000000
[[1,2,5,7],[3,8],[4],[6]]=>[6,4,3,8,1,2,5,7]=>[5,3,2,6,1,7,8,4]=>[5,6,3,2,7,1,8,4]=>0000000
[[1,2,3,7],[4,8],[5],[6]]=>[6,5,4,8,1,2,3,7]=>[5,6,3,2,1,7,8,4]=>[5,6,7,3,2,1,8,4]=>0000000
[[1,2,4,6],[3,8],[5],[7]]=>[7,5,3,8,1,2,4,6]=>[5,3,6,2,7,1,8,4]=>[5,6,3,7,2,8,1,4]=>0000000
[[1,3,4,5],[2,8],[6],[7]]=>[7,6,2,8,1,3,4,5]=>[3,5,6,7,2,1,8,4]=>[5,3,6,7,8,2,1,4]=>0000000
[[1,2,3,5],[4,8],[6],[7]]=>[7,6,4,8,1,2,3,5]=>[5,6,3,7,2,1,8,4]=>[5,6,7,3,8,2,1,4]=>0000000
[[1,2,3,4],[5,8],[6],[7]]=>[7,6,5,8,1,2,3,4]=>[5,6,7,3,2,1,8,4]=>[5,6,7,8,3,2,1,4]=>0000000
[[1,2,4,7],[3,6],[5],[8]]=>[8,5,3,6,1,2,4,7]=>[5,3,6,2,7,4,8,1]=>[5,6,3,7,2,4,8,1]=>0000000
[[1,2,5,7],[3,4],[6],[8]]=>[8,6,3,4,1,2,5,7]=>[5,3,6,4,7,2,8,1]=>[5,6,3,4,7,2,8,1]=>0000000
[[1,3,4,7],[2,5],[6],[8]]=>[8,6,2,5,1,3,4,7]=>[3,5,6,7,4,2,8,1]=>[5,3,6,7,4,2,8,1]=>0000000
[[1,3,4,6],[2,7],[5],[8]]=>[8,5,2,7,1,3,4,6]=>[3,5,6,2,7,8,4,1]=>[5,3,6,7,2,8,4,1]=>0000000
[[1,2,3,6],[4,7],[5],[8]]=>[8,5,4,7,1,2,3,6]=>[5,6,3,2,7,8,4,1]=>[5,6,7,3,2,8,4,1]=>0000000
[[1,2,4,5],[3,7],[6],[8]]=>[8,6,3,7,1,2,4,5]=>[5,3,6,7,2,8,4,1]=>[5,6,3,7,8,2,4,1]=>0000000
[[1,2,3,4],[5,7],[6],[8]]=>[8,6,5,7,1,2,3,4]=>[5,6,7,3,2,8,4,1]=>[5,6,7,8,3,2,4,1]=>0000000
[[1,2,4,6],[3,5],[7],[8]]=>[8,7,3,5,1,2,4,6]=>[5,3,6,7,4,8,2,1]=>[5,6,3,7,4,8,2,1]=>0000000
[[1,3,4,5],[2,6],[7],[8]]=>[8,7,2,6,1,3,4,5]=>[3,5,6,7,8,4,2,1]=>[5,3,6,7,8,4,2,1]=>0000000
[[1,2,4,5],[3,6],[7],[8]]=>[8,7,3,6,1,2,4,5]=>[5,3,6,7,8,4,2,1]=>[5,6,3,7,8,4,2,1]=>0000000
[[1,2,3,5],[4,6],[7],[8]]=>[8,7,4,6,1,2,3,5]=>[5,6,3,7,8,4,2,1]=>[5,6,7,3,8,4,2,1]=>0000000
[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[5,6,7,3,8,4,2,1]=>[5,6,7,8,3,4,2,1]=>0000000
[[1,6,7,8],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7,8]=>[5,4,3,2,1,6,7,8]=>[5,4,3,2,1,6,7,8]=>0000111
[[1,5,7,8],[2],[3],[4],[6]]=>[6,4,3,2,1,5,7,8]=>[5,4,3,2,6,1,7,8]=>[5,4,3,2,6,1,7,8]=>0000011
[[1,5,6,8],[2],[3],[4],[7]]=>[7,4,3,2,1,5,6,8]=>[5,4,3,2,6,7,1,8]=>[5,4,3,2,6,7,1,8]=>0000001
[[1,4,6,8],[2],[3],[5],[7]]=>[7,5,3,2,1,4,6,8]=>[5,4,3,6,2,7,1,8]=>[5,4,3,6,2,7,1,8]=>0000001
[[1,4,5,8],[2],[3],[6],[7]]=>[7,6,3,2,1,4,5,8]=>[5,4,3,6,7,2,1,8]=>[5,4,3,6,7,2,1,8]=>0000001
[[1,3,5,8],[2],[4],[6],[7]]=>[7,6,4,2,1,3,5,8]=>[5,4,6,3,7,2,1,8]=>[5,4,6,3,7,2,1,8]=>0000001
[[1,2,3,8],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3,8]=>[5,6,7,4,3,2,1,8]=>[5,6,7,4,3,2,1,8]=>0000001
[[1,5,6,7],[2],[3],[4],[8]]=>[8,4,3,2,1,5,6,7]=>[5,4,3,2,6,7,8,1]=>[5,4,3,2,6,7,8,1]=>0000000
[[1,4,6,7],[2],[3],[5],[8]]=>[8,5,3,2,1,4,6,7]=>[5,4,3,6,2,7,8,1]=>[5,4,3,6,2,7,8,1]=>0000000
[[1,3,6,7],[2],[4],[5],[8]]=>[8,5,4,2,1,3,6,7]=>[5,4,6,3,2,7,8,1]=>[5,4,6,3,2,7,8,1]=>0000000
[[1,4,5,7],[2],[3],[6],[8]]=>[8,6,3,2,1,4,5,7]=>[5,4,3,6,7,2,8,1]=>[5,4,3,6,7,2,8,1]=>0000000
[[1,3,5,7],[2],[4],[6],[8]]=>[8,6,4,2,1,3,5,7]=>[5,4,6,3,7,2,8,1]=>[5,4,6,3,7,2,8,1]=>0000000
[[1,3,4,6],[2],[5],[7],[8]]=>[8,7,5,2,1,3,4,6]=>[5,4,6,7,3,8,2,1]=>[5,4,6,7,3,8,2,1]=>0000000
[[1,3,4,5],[2],[6],[7],[8]]=>[8,7,6,2,1,3,4,5]=>[5,4,6,7,8,3,2,1]=>[5,4,6,7,8,3,2,1]=>0000000
[[1,2,4,5],[3],[6],[7],[8]]=>[8,7,6,3,1,2,4,5]=>[5,6,4,7,8,3,2,1]=>[5,6,4,7,8,3,2,1]=>0000000
[[1,2,3,5],[4],[6],[7],[8]]=>[8,7,6,4,1,2,3,5]=>[5,6,7,4,8,3,2,1]=>[5,6,7,4,8,3,2,1]=>0000000
[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[5,6,7,8,4,3,2,1]=>[5,6,7,8,4,3,2,1]=>0000000
[[1,2,6],[3,4,8],[5,7]]=>[5,7,3,4,8,1,2,6]=>[3,1,6,4,7,2,8,5]=>[6,7,3,4,1,8,2,5]=>0000000
[[1,4,5],[2,6,8],[3,7]]=>[3,7,2,6,8,1,4,5]=>[6,7,1,3,4,2,8,5]=>[6,3,1,7,8,4,2,5]=>0000000
[[1,2,5],[3,6,8],[4,7]]=>[4,7,3,6,8,1,2,5]=>[1,3,6,7,4,2,8,5]=>[6,7,3,1,8,4,2,5]=>0000000
[[1,2,3],[4,6,8],[5,7]]=>[5,7,4,6,8,1,2,3]=>[1,6,3,7,4,2,8,5]=>[6,7,8,3,1,4,2,5]=>0000000
[[1,2,4],[3,5,8],[6,7]]=>[6,7,3,5,8,1,2,4]=>[3,6,1,7,4,2,8,5]=>[6,7,3,8,4,1,2,5]=>0000000
[[1,3,6],[2,5,7],[4,8]]=>[4,8,2,5,7,1,3,6]=>[1,6,3,4,7,8,5,2]=>[6,7,1,3,4,8,5,2]=>0000000
[[1,2,5],[3,6,7],[4,8]]=>[4,8,3,6,7,1,2,5]=>[1,3,6,7,4,8,5,2]=>[6,7,3,1,8,4,5,2]=>0000000
[[1,2,3],[4,6,7],[5,8]]=>[5,8,4,6,7,1,2,3]=>[1,6,3,7,4,8,5,2]=>[6,7,8,3,1,4,5,2]=>0000000
[[1,2,4],[3,5,7],[6,8]]=>[6,8,3,5,7,1,2,4]=>[3,6,1,7,4,8,5,2]=>[6,7,3,8,4,1,5,2]=>0000000
[[1,3,5],[2,4,6],[7,8]]=>[7,8,2,4,6,1,3,5]=>[6,3,4,7,1,8,5,2]=>[6,3,7,4,8,5,1,2]=>0000000
[[1,2,4],[3,5,6],[7,8]]=>[7,8,3,5,6,1,2,4]=>[3,6,7,4,1,8,5,2]=>[6,7,3,8,4,5,1,2]=>0000000
[[1,5,7],[2,6,8],[3],[4]]=>[4,3,2,6,8,1,5,7]=>[2,1,6,3,4,7,8,5]=>[6,2,7,1,3,4,8,5]=>0000000
[[1,3,7],[2,6,8],[4],[5]]=>[5,4,2,6,8,1,3,7]=>[6,3,2,1,4,7,8,5]=>[6,3,7,2,1,4,8,5]=>0000000
[[1,2,7],[3,6,8],[4],[5]]=>[5,4,3,6,8,1,2,7]=>[3,2,1,6,4,7,8,5]=>[6,7,3,2,1,4,8,5]=>0000000
[[1,4,7],[2,5,8],[3],[6]]=>[6,3,2,5,8,1,4,7]=>[2,6,3,4,1,7,8,5]=>[6,2,7,3,4,1,8,5]=>0000000
[[1,3,6],[2,7,8],[4],[5]]=>[5,4,2,7,8,1,3,6]=>[6,2,1,7,3,4,8,5]=>[6,2,7,8,1,3,4,5]=>0000000
[[1,2,6],[3,7,8],[4],[5]]=>[5,4,3,7,8,1,2,6]=>[3,2,1,6,7,4,8,5]=>[6,7,3,2,1,8,4,5]=>0000000
[[1,3,5],[2,7,8],[4],[6]]=>[6,4,2,7,8,1,3,5]=>[6,2,7,3,1,4,8,5]=>[6,2,7,8,3,1,4,5]=>0000000
[[1,2,5],[3,7,8],[4],[6]]=>[6,4,3,7,8,1,2,5]=>[3,2,6,1,7,4,8,5]=>[6,7,3,2,8,1,4,5]=>0000000
[[1,2,3],[4,7,8],[5],[6]]=>[6,5,4,7,8,1,2,3]=>[6,3,2,1,7,4,8,5]=>[6,7,8,3,2,1,4,5]=>0000000
[[1,4,5],[2,6,8],[3],[7]]=>[7,3,2,6,8,1,4,5]=>[2,6,7,3,4,1,8,5]=>[6,2,7,3,8,4,1,5]=>0000000
[[1,2,4],[3,6,8],[5],[7]]=>[7,5,3,6,8,1,2,4]=>[3,6,2,7,4,1,8,5]=>[6,7,3,8,2,4,1,5]=>0000000
[[1,2,3],[4,6,8],[5],[7]]=>[7,5,4,6,8,1,2,3]=>[6,3,2,7,4,1,8,5]=>[6,7,8,3,2,4,1,5]=>0000000
[[1,2,3],[4,5,8],[6],[7]]=>[7,6,4,5,8,1,2,3]=>[6,3,7,4,2,1,8,5]=>[6,7,8,3,4,2,1,5]=>0000000
[[1,2,4],[3,6,7],[5],[8]]=>[8,5,3,6,7,1,2,4]=>[3,6,2,7,4,8,5,1]=>[6,7,3,8,2,4,5,1]=>0000000
[[1,3,4],[2,5,6],[7],[8]]=>[8,7,2,5,6,1,3,4]=>[6,7,3,4,8,5,2,1]=>[6,3,7,8,4,5,2,1]=>0000000
[[1,2,4],[3,5,6],[7],[8]]=>[8,7,3,5,6,1,2,4]=>[3,6,7,4,8,5,2,1]=>[6,7,3,8,4,5,2,1]=>0000000
[[1,5,8],[2,6],[3,7],[4]]=>[4,3,7,2,6,1,5,8]=>[2,1,4,6,7,5,3,8]=>[6,4,2,1,7,5,3,8]=>0000001
[[1,3,8],[2,6],[4,7],[5]]=>[5,4,7,2,6,1,3,8]=>[4,2,1,6,7,5,3,8]=>[6,4,7,2,1,5,3,8]=>0000001
[[1,3,8],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3,8]=>[4,2,6,1,7,5,3,8]=>[6,4,7,2,5,1,3,8]=>0000001
[[1,4,8],[2,5],[3,6],[7]]=>[7,3,6,2,5,1,4,8]=>[2,4,6,7,5,3,1,8]=>[6,4,2,7,5,3,1,8]=>0000001
[[1,2,8],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2,8]=>[6,4,2,7,5,3,1,8]=>[6,7,4,5,2,3,1,8]=>0000001
[[1,5,7],[2,6],[3,8],[4]]=>[4,3,8,2,6,1,5,7]=>[2,1,4,6,7,5,8,3]=>[6,4,2,1,7,5,8,3]=>0000000
[[1,3,7],[2,5],[4,8],[6]]=>[6,4,8,2,5,1,3,7]=>[4,2,6,1,7,5,8,3]=>[6,4,7,2,5,1,8,3]=>0000000
[[1,3,5],[2,7],[4,8],[6]]=>[6,4,8,2,7,1,3,5]=>[4,2,6,1,7,8,5,3]=>[6,4,7,2,8,1,5,3]=>0000000
[[1,3,4],[2,7],[5,8],[6]]=>[6,5,8,2,7,1,3,4]=>[4,6,2,1,7,8,5,3]=>[6,4,7,8,2,1,5,3]=>0000000
[[1,4,6],[2,5],[3,8],[7]]=>[7,3,8,2,5,1,4,6]=>[2,4,6,7,5,1,8,3]=>[6,4,2,7,5,8,1,3]=>0000000
[[1,2,6],[3,4],[5,8],[7]]=>[7,5,8,3,4,1,2,6]=>[6,4,2,7,5,1,8,3]=>[6,7,4,5,2,8,1,3]=>0000000
[[1,3,5],[2,6],[4,8],[7]]=>[7,4,8,2,6,1,3,5]=>[4,2,6,7,1,8,5,3]=>[6,4,7,2,8,5,1,3]=>0000000
[[1,2,4],[3,6],[5,8],[7]]=>[7,5,8,3,6,1,2,4]=>[6,4,2,7,1,8,5,3]=>[6,7,4,8,2,5,1,3]=>0000000
[[1,2,3],[4,6],[5,8],[7]]=>[7,5,8,4,6,1,2,3]=>[6,2,7,4,1,8,5,3]=>[6,7,8,4,2,5,1,3]=>0000000
[[1,3,7],[2,5],[4,6],[8]]=>[8,4,6,2,5,1,3,7]=>[4,2,6,7,5,3,8,1]=>[6,4,7,2,5,3,8,1]=>0000000
[[1,3,4],[2,6],[5,7],[8]]=>[8,5,7,2,6,1,3,4]=>[4,6,2,7,8,5,3,1]=>[6,4,7,8,2,5,3,1]=>0000000
[[1,3,5],[2,4],[6,7],[8]]=>[8,6,7,2,4,1,3,5]=>[4,6,7,5,2,8,3,1]=>[6,4,7,5,8,2,3,1]=>0000000
[[1,3,4],[2,5],[6,7],[8]]=>[8,6,7,2,5,1,3,4]=>[4,6,7,2,8,5,3,1]=>[6,4,7,8,5,2,3,1]=>0000000
[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>[6,7,4,2,8,5,3,1]=>[6,7,8,4,5,2,3,1]=>0000000
[[1,6,8],[2,7],[3],[4],[5]]=>[5,4,3,2,7,1,6,8]=>[4,3,2,1,6,7,5,8]=>[6,4,3,2,1,7,5,8]=>0000001
[[1,4,8],[2,7],[3],[5],[6]]=>[6,5,3,2,7,1,4,8]=>[4,3,6,2,1,7,5,8]=>[6,4,3,7,2,1,5,8]=>0000001
[[1,4,8],[2,6],[3],[5],[7]]=>[7,5,3,2,6,1,4,8]=>[4,3,6,2,7,5,1,8]=>[6,4,3,7,2,5,1,8]=>0000001
[[1,6,7],[2,8],[3],[4],[5]]=>[5,4,3,2,8,1,6,7]=>[4,3,2,1,6,7,8,5]=>[6,4,3,2,1,7,8,5]=>0000000
[[1,4,7],[2,8],[3],[5],[6]]=>[6,5,3,2,8,1,4,7]=>[4,3,6,2,1,7,8,5]=>[6,4,3,7,2,1,8,5]=>0000000
[[1,4,6],[2,8],[3],[5],[7]]=>[7,5,3,2,8,1,4,6]=>[4,3,6,2,7,1,8,5]=>[6,4,3,7,2,8,1,5]=>0000000
[[1,2,3],[4,8],[5],[6],[7]]=>[7,6,5,4,8,1,2,3]=>[6,7,4,3,2,1,8,5]=>[6,7,8,4,3,2,1,5]=>0000000
[[1,4,7],[2,6],[3],[5],[8]]=>[8,5,3,2,6,1,4,7]=>[4,3,6,2,7,5,8,1]=>[6,4,3,7,2,5,8,1]=>0000000
[[1,3,7],[2,6],[4],[5],[8]]=>[8,5,4,2,6,1,3,7]=>[4,6,3,2,7,5,8,1]=>[6,4,7,3,2,5,8,1]=>0000000
[[1,4,6],[2,7],[3],[5],[8]]=>[8,5,3,2,7,1,4,6]=>[4,3,6,2,7,8,5,1]=>[6,4,3,7,2,8,5,1]=>0000000
[[1,4,5],[2,6],[3],[7],[8]]=>[8,7,3,2,6,1,4,5]=>[4,3,6,7,8,5,2,1]=>[6,4,3,7,8,5,2,1]=>0000000
[[1,3,5],[2,6],[4],[7],[8]]=>[8,7,4,2,6,1,3,5]=>[4,6,3,7,8,5,2,1]=>[6,4,7,3,8,5,2,1]=>0000000
[[1,3,4],[2,6],[5],[7],[8]]=>[8,7,5,2,6,1,3,4]=>[4,6,7,3,8,5,2,1]=>[6,4,7,8,3,5,2,1]=>0000000
[[1,2,4],[3,6],[5],[7],[8]]=>[8,7,5,3,6,1,2,4]=>[6,4,7,3,8,5,2,1]=>[6,7,4,8,3,5,2,1]=>0000000
[[1,3,5],[2,4],[6],[7],[8]]=>[8,7,6,2,4,1,3,5]=>[4,6,7,5,8,3,2,1]=>[6,4,7,5,8,3,2,1]=>0000000
[[1,2,5],[3,4],[6],[7],[8]]=>[8,7,6,3,4,1,2,5]=>[6,4,7,5,8,3,2,1]=>[6,7,4,5,8,3,2,1]=>0000000
[[1,5,8],[2],[3],[4],[6],[7]]=>[7,6,4,3,2,1,5,8]=>[6,5,4,3,7,2,1,8]=>[6,5,4,3,7,2,1,8]=>0000001
[[1,5,7],[2],[3],[4],[6],[8]]=>[8,6,4,3,2,1,5,7]=>[6,5,4,3,7,2,8,1]=>[6,5,4,3,7,2,8,1]=>0000000
[[1,5,6],[2],[3],[4],[7],[8]]=>[8,7,4,3,2,1,5,6]=>[6,5,4,3,7,8,2,1]=>[6,5,4,3,7,8,2,1]=>0000000
[[1,4,5],[2],[3],[6],[7],[8]]=>[8,7,6,3,2,1,4,5]=>[6,5,4,7,8,3,2,1]=>[6,5,4,7,8,3,2,1]=>0000000
[[1,3,5],[2],[4],[6],[7],[8]]=>[8,7,6,4,2,1,3,5]=>[6,5,7,4,8,3,2,1]=>[6,5,7,4,8,3,2,1]=>0000000
[[1,2,5],[3],[4],[6],[7],[8]]=>[8,7,6,4,3,1,2,5]=>[6,7,5,4,8,3,2,1]=>[6,7,5,4,8,3,2,1]=>0000000
[[1,2,4],[3],[5],[6],[7],[8]]=>[8,7,6,5,3,1,2,4]=>[6,7,5,8,4,3,2,1]=>[6,7,5,8,4,3,2,1]=>0000000
[[1,5],[2,6],[3,7],[4,8]]=>[4,8,3,7,2,6,1,5]=>[1,3,5,7,8,6,4,2]=>[7,5,3,1,8,6,4,2]=>0000000
[[1,2],[3,6],[4,7],[5,8]]=>[5,8,4,7,3,6,1,2]=>[1,3,7,5,8,6,4,2]=>[7,8,5,3,1,6,4,2]=>0000000
[[1,4],[2,5],[3,7],[6,8]]=>[6,8,3,7,2,5,1,4]=>[3,5,1,7,8,6,4,2]=>[7,5,3,8,6,1,4,2]=>0000000
[[1,3],[2,4],[5,7],[6,8]]=>[6,8,5,7,2,4,1,3]=>[5,1,7,3,8,6,4,2]=>[7,5,8,6,3,1,4,2]=>0000000
[[1,2],[3,4],[5,7],[6,8]]=>[6,8,5,7,3,4,1,2]=>[1,7,5,3,8,6,4,2]=>[7,8,5,6,3,1,4,2]=>0000000
[[1,2],[3,5],[4,6],[7,8]]=>[7,8,4,6,3,5,1,2]=>[3,7,5,1,8,6,4,2]=>[7,8,5,3,6,4,1,2]=>0000000
[[1,3],[2,4],[5,6],[7,8]]=>[7,8,5,6,2,4,1,3]=>[5,7,3,1,8,6,4,2]=>[7,5,8,6,3,4,1,2]=>0000000
[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[7,5,3,1,8,6,4,2]=>[7,8,5,6,3,4,1,2]=>0000000
[[1,5],[2,7],[3,8],[4],[6]]=>[6,4,3,8,2,7,1,5]=>[3,2,5,1,7,8,6,4]=>[7,5,3,2,8,1,6,4]=>0000000
[[1,3],[2,7],[4,8],[5],[6]]=>[6,5,4,8,2,7,1,3]=>[5,3,2,1,7,8,6,4]=>[7,5,8,3,2,1,6,4]=>0000000
[[1,5],[2,6],[3,8],[4],[7]]=>[7,4,3,8,2,6,1,5]=>[3,2,5,7,1,8,6,4]=>[7,5,3,2,8,6,1,4]=>0000000
[[1,4],[2,6],[3,8],[5],[7]]=>[7,5,3,8,2,6,1,4]=>[3,5,2,7,1,8,6,4]=>[7,5,3,8,2,6,1,4]=>0000000
[[1,3],[2,6],[4,8],[5],[7]]=>[7,5,4,8,2,6,1,3]=>[5,3,2,7,1,8,6,4]=>[7,5,8,3,2,6,1,4]=>0000000
[[1,3],[2,5],[4,8],[6],[7]]=>[7,6,4,8,2,5,1,3]=>[5,3,7,2,1,8,6,4]=>[7,5,8,3,6,2,1,4]=>0000000
[[1,2],[3,4],[5,8],[6],[7]]=>[7,6,5,8,3,4,1,2]=>[7,5,3,2,1,8,6,4]=>[7,8,5,6,3,2,1,4]=>0000000
[[1,4],[2,5],[3,7],[6],[8]]=>[8,6,3,7,2,5,1,4]=>[3,5,7,2,8,6,4,1]=>[7,5,3,8,6,2,4,1]=>0000000
[[1,3],[2,5],[4,7],[6],[8]]=>[8,6,4,7,2,5,1,3]=>[5,3,7,2,8,6,4,1]=>[7,5,8,3,6,2,4,1]=>0000000
[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>[7,5,3,8,6,4,2,1]=>[7,8,5,6,3,4,2,1]=>0000000
[[1,6],[2,8],[3],[4],[5],[7]]=>[7,5,4,3,2,8,1,6]=>[5,4,3,2,7,1,8,6]=>[7,5,4,3,2,8,1,6]=>0000000
[[1,5],[2,8],[3],[4],[6],[7]]=>[7,6,4,3,2,8,1,5]=>[5,4,3,7,2,1,8,6]=>[7,5,4,3,8,2,1,6]=>0000000
[[1,2],[3,8],[4],[5],[6],[7]]=>[7,6,5,4,3,8,1,2]=>[7,5,4,3,2,1,8,6]=>[7,8,5,4,3,2,1,6]=>0000000
[[1,8],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1,8]=>[7,6,5,4,3,2,1,8]=>[7,6,5,4,3,2,1,8]=>0000001
[[1,5],[2],[3],[4],[6],[7],[8]]=>[8,7,6,4,3,2,1,5]=>[7,6,5,4,8,3,2,1]=>[7,6,5,4,8,3,2,1]=>0000000
[[1,4],[2],[3],[5],[6],[7],[8]]=>[8,7,6,5,3,2,1,4]=>[7,6,5,8,4,3,2,1]=>[7,6,5,8,4,3,2,1]=>0000000
[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>0000000
[[1,2,3,4,5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,1,2,3,4,5]=>[6,7,8,9,10,5,4,3,2,1]=>[6,7,8,9,10,5,4,3,2,1]=>000000000
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
Lehmer-code to major-code bijection
Description
Sends a permutation to the unique permutation such that the Lehmer code is sent to the major code.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
- If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
- If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
- $1$
- $|1|4 \to 14$
- $|14|2 \to 412$
- $|4|1|2|5 \to 4125$
- $|4|125|3 \to 45123.$
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
Map
connectivity set
Description
The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.
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