Identifier
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Images
=>
Cc0007;cc-rep-0
[[1]]=>[1]=>[1]=>[] [[1,2]]=>[1,2]=>[1,2]=>[1] [[1],[2]]=>[2,1]=>[2,1]=>[1] [[1,2,3]]=>[1,2,3]=>[1,2,3]=>[1,2] [[1,3],[2]]=>[2,1,3]=>[2,1,3]=>[2,1] [[1,2],[3]]=>[3,1,2]=>[1,3,2]=>[1,2] [[1],[2],[3]]=>[3,2,1]=>[3,2,1]=>[2,1] [[1,2,3,4]]=>[1,2,3,4]=>[1,2,3,4]=>[1,2,3] [[1,3,4],[2]]=>[2,1,3,4]=>[2,1,3,4]=>[2,1,3] [[1,2,4],[3]]=>[3,1,2,4]=>[1,3,2,4]=>[1,3,2] [[1,2,3],[4]]=>[4,1,2,3]=>[1,2,4,3]=>[1,2,3] [[1,3],[2,4]]=>[2,4,1,3]=>[2,1,4,3]=>[2,1,3] [[1,2],[3,4]]=>[3,4,1,2]=>[1,3,4,2]=>[1,3,2] [[1,4],[2],[3]]=>[3,2,1,4]=>[3,2,1,4]=>[3,2,1] [[1,3],[2],[4]]=>[4,2,1,3]=>[2,4,1,3]=>[2,1,3] [[1,2],[3],[4]]=>[4,3,1,2]=>[1,4,3,2]=>[1,3,2] [[1],[2],[3],[4]]=>[4,3,2,1]=>[4,3,2,1]=>[3,2,1] [[1,2,3,4,5]]=>[1,2,3,4,5]=>[1,2,3,4,5]=>[1,2,3,4] [[1,3,4,5],[2]]=>[2,1,3,4,5]=>[2,1,3,4,5]=>[2,1,3,4] [[1,2,4,5],[3]]=>[3,1,2,4,5]=>[1,3,2,4,5]=>[1,3,2,4] [[1,2,3,5],[4]]=>[4,1,2,3,5]=>[1,2,4,3,5]=>[1,2,4,3] [[1,2,3,4],[5]]=>[5,1,2,3,4]=>[1,2,3,5,4]=>[1,2,3,4] [[1,3,5],[2,4]]=>[2,4,1,3,5]=>[2,1,4,3,5]=>[2,1,4,3] [[1,2,5],[3,4]]=>[3,4,1,2,5]=>[1,3,4,2,5]=>[1,3,4,2] [[1,3,4],[2,5]]=>[2,5,1,3,4]=>[2,1,3,5,4]=>[2,1,3,4] [[1,2,4],[3,5]]=>[3,5,1,2,4]=>[1,3,2,5,4]=>[1,3,2,4] [[1,2,3],[4,5]]=>[4,5,1,2,3]=>[1,2,4,5,3]=>[1,2,4,3] [[1,4,5],[2],[3]]=>[3,2,1,4,5]=>[3,2,1,4,5]=>[3,2,1,4] [[1,3,5],[2],[4]]=>[4,2,1,3,5]=>[2,4,1,3,5]=>[2,4,1,3] [[1,2,5],[3],[4]]=>[4,3,1,2,5]=>[1,4,3,2,5]=>[1,4,3,2] [[1,3,4],[2],[5]]=>[5,2,1,3,4]=>[2,1,5,3,4]=>[2,1,3,4] [[1,2,4],[3],[5]]=>[5,3,1,2,4]=>[1,3,5,2,4]=>[1,3,2,4] [[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[1,2,5,4,3]=>[1,2,4,3] [[1,4],[2,5],[3]]=>[3,2,5,1,4]=>[3,2,1,5,4]=>[3,2,1,4] [[1,3],[2,5],[4]]=>[4,2,5,1,3]=>[2,4,1,5,3]=>[2,4,1,3] [[1,2],[3,5],[4]]=>[4,3,5,1,2]=>[1,4,3,5,2]=>[1,4,3,2] [[1,3],[2,4],[5]]=>[5,2,4,1,3]=>[2,1,5,4,3]=>[2,1,4,3] [[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[1,3,5,4,2]=>[1,3,4,2] [[1,5],[2],[3],[4]]=>[4,3,2,1,5]=>[4,3,2,1,5]=>[4,3,2,1] [[1,4],[2],[3],[5]]=>[5,3,2,1,4]=>[3,5,2,1,4]=>[3,2,1,4] [[1,3],[2],[4],[5]]=>[5,4,2,1,3]=>[2,5,4,1,3]=>[2,4,1,3] [[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[1,5,4,3,2]=>[1,4,3,2] [[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[5,4,3,2,1]=>[4,3,2,1] [[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>[1,2,3,4,5] [[1,3,4,5,6],[2]]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>[2,1,3,4,5] [[1,2,4,5,6],[3]]=>[3,1,2,4,5,6]=>[1,3,2,4,5,6]=>[1,3,2,4,5] [[1,2,3,5,6],[4]]=>[4,1,2,3,5,6]=>[1,2,4,3,5,6]=>[1,2,4,3,5] [[1,2,3,4,6],[5]]=>[5,1,2,3,4,6]=>[1,2,3,5,4,6]=>[1,2,3,5,4] [[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[1,2,3,4,6,5]=>[1,2,3,4,5] [[1,3,5,6],[2,4]]=>[2,4,1,3,5,6]=>[2,1,4,3,5,6]=>[2,1,4,3,5] [[1,2,5,6],[3,4]]=>[3,4,1,2,5,6]=>[1,3,4,2,5,6]=>[1,3,4,2,5] [[1,3,4,6],[2,5]]=>[2,5,1,3,4,6]=>[2,1,3,5,4,6]=>[2,1,3,5,4] [[1,2,4,6],[3,5]]=>[3,5,1,2,4,6]=>[1,3,2,5,4,6]=>[1,3,2,5,4] [[1,2,3,6],[4,5]]=>[4,5,1,2,3,6]=>[1,2,4,5,3,6]=>[1,2,4,5,3] [[1,3,4,5],[2,6]]=>[2,6,1,3,4,5]=>[2,1,3,4,6,5]=>[2,1,3,4,5] [[1,2,4,5],[3,6]]=>[3,6,1,2,4,5]=>[1,3,2,4,6,5]=>[1,3,2,4,5] [[1,2,3,5],[4,6]]=>[4,6,1,2,3,5]=>[1,2,4,3,6,5]=>[1,2,4,3,5] [[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[1,2,3,5,6,4]=>[1,2,3,5,4] [[1,4,5,6],[2],[3]]=>[3,2,1,4,5,6]=>[3,2,1,4,5,6]=>[3,2,1,4,5] [[1,3,5,6],[2],[4]]=>[4,2,1,3,5,6]=>[2,4,1,3,5,6]=>[2,4,1,3,5] [[1,2,5,6],[3],[4]]=>[4,3,1,2,5,6]=>[1,4,3,2,5,6]=>[1,4,3,2,5] [[1,3,4,6],[2],[5]]=>[5,2,1,3,4,6]=>[2,1,5,3,4,6]=>[2,1,5,3,4] [[1,2,4,6],[3],[5]]=>[5,3,1,2,4,6]=>[1,3,5,2,4,6]=>[1,3,5,2,4] [[1,2,3,6],[4],[5]]=>[5,4,1,2,3,6]=>[1,2,5,4,3,6]=>[1,2,5,4,3] [[1,3,4,5],[2],[6]]=>[6,2,1,3,4,5]=>[2,1,3,6,4,5]=>[2,1,3,4,5] [[1,2,4,5],[3],[6]]=>[6,3,1,2,4,5]=>[1,3,2,6,4,5]=>[1,3,2,4,5] [[1,2,3,5],[4],[6]]=>[6,4,1,2,3,5]=>[1,2,4,6,3,5]=>[1,2,4,3,5] [[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[1,2,3,6,5,4]=>[1,2,3,5,4] [[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[2,1,4,3,6,5]=>[2,1,4,3,5] [[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[1,3,4,2,6,5]=>[1,3,4,2,5] [[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[2,1,3,5,6,4]=>[2,1,3,5,4] [[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[1,3,2,5,6,4]=>[1,3,2,5,4] [[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[1,2,4,5,6,3]=>[1,2,4,5,3] [[1,4,6],[2,5],[3]]=>[3,2,5,1,4,6]=>[3,2,1,5,4,6]=>[3,2,1,5,4] [[1,3,6],[2,5],[4]]=>[4,2,5,1,3,6]=>[2,4,1,5,3,6]=>[2,4,1,5,3] [[1,2,6],[3,5],[4]]=>[4,3,5,1,2,6]=>[1,4,3,5,2,6]=>[1,4,3,5,2] [[1,3,6],[2,4],[5]]=>[5,2,4,1,3,6]=>[2,1,5,4,3,6]=>[2,1,5,4,3] [[1,2,6],[3,4],[5]]=>[5,3,4,1,2,6]=>[1,3,5,4,2,6]=>[1,3,5,4,2] [[1,4,5],[2,6],[3]]=>[3,2,6,1,4,5]=>[3,2,1,4,6,5]=>[3,2,1,4,5] [[1,3,5],[2,6],[4]]=>[4,2,6,1,3,5]=>[2,4,1,3,6,5]=>[2,4,1,3,5] [[1,2,5],[3,6],[4]]=>[4,3,6,1,2,5]=>[1,4,3,2,6,5]=>[1,4,3,2,5] [[1,3,4],[2,6],[5]]=>[5,2,6,1,3,4]=>[2,1,5,3,6,4]=>[2,1,5,3,4] [[1,2,4],[3,6],[5]]=>[5,3,6,1,2,4]=>[1,3,5,2,6,4]=>[1,3,5,2,4] [[1,2,3],[4,6],[5]]=>[5,4,6,1,2,3]=>[1,2,5,4,6,3]=>[1,2,5,4,3] [[1,3,5],[2,4],[6]]=>[6,2,4,1,3,5]=>[2,1,4,6,3,5]=>[2,1,4,3,5] [[1,2,5],[3,4],[6]]=>[6,3,4,1,2,5]=>[1,3,4,6,2,5]=>[1,3,4,2,5] [[1,3,4],[2,5],[6]]=>[6,2,5,1,3,4]=>[2,1,3,6,5,4]=>[2,1,3,5,4] [[1,2,4],[3,5],[6]]=>[6,3,5,1,2,4]=>[1,3,2,6,5,4]=>[1,3,2,5,4] [[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[1,2,4,6,5,3]=>[1,2,4,5,3] [[1,5,6],[2],[3],[4]]=>[4,3,2,1,5,6]=>[4,3,2,1,5,6]=>[4,3,2,1,5] [[1,4,6],[2],[3],[5]]=>[5,3,2,1,4,6]=>[3,5,2,1,4,6]=>[3,5,2,1,4] [[1,3,6],[2],[4],[5]]=>[5,4,2,1,3,6]=>[2,5,4,1,3,6]=>[2,5,4,1,3] [[1,2,6],[3],[4],[5]]=>[5,4,3,1,2,6]=>[1,5,4,3,2,6]=>[1,5,4,3,2] [[1,4,5],[2],[3],[6]]=>[6,3,2,1,4,5]=>[3,2,6,1,4,5]=>[3,2,1,4,5] [[1,3,5],[2],[4],[6]]=>[6,4,2,1,3,5]=>[2,4,6,1,3,5]=>[2,4,1,3,5] [[1,2,5],[3],[4],[6]]=>[6,4,3,1,2,5]=>[1,4,6,3,2,5]=>[1,4,3,2,5] [[1,3,4],[2],[5],[6]]=>[6,5,2,1,3,4]=>[2,1,6,5,3,4]=>[2,1,5,3,4] [[1,2,4],[3],[5],[6]]=>[6,5,3,1,2,4]=>[1,3,6,5,2,4]=>[1,3,5,2,4] [[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[1,2,6,5,4,3]=>[1,2,5,4,3] [[1,4],[2,5],[3,6]]=>[3,6,2,5,1,4]=>[3,2,1,6,5,4]=>[3,2,1,5,4] [[1,3],[2,5],[4,6]]=>[4,6,2,5,1,3]=>[2,4,1,6,5,3]=>[2,4,1,5,3] [[1,2],[3,5],[4,6]]=>[4,6,3,5,1,2]=>[1,4,3,6,5,2]=>[1,4,3,5,2] [[1,3],[2,4],[5,6]]=>[5,6,2,4,1,3]=>[2,1,5,6,4,3]=>[2,1,5,4,3] [[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[1,3,5,6,4,2]=>[1,3,5,4,2] [[1,5],[2,6],[3],[4]]=>[4,3,2,6,1,5]=>[4,3,2,1,6,5]=>[4,3,2,1,5] [[1,4],[2,6],[3],[5]]=>[5,3,2,6,1,4]=>[3,5,2,1,6,4]=>[3,5,2,1,4] [[1,3],[2,6],[4],[5]]=>[5,4,2,6,1,3]=>[2,5,4,1,6,3]=>[2,5,4,1,3] [[1,2],[3,6],[4],[5]]=>[5,4,3,6,1,2]=>[1,5,4,3,6,2]=>[1,5,4,3,2] [[1,4],[2,5],[3],[6]]=>[6,3,2,5,1,4]=>[3,2,6,1,5,4]=>[3,2,1,5,4] [[1,3],[2,5],[4],[6]]=>[6,4,2,5,1,3]=>[2,4,6,1,5,3]=>[2,4,1,5,3] [[1,2],[3,5],[4],[6]]=>[6,4,3,5,1,2]=>[1,4,6,3,5,2]=>[1,4,3,5,2] [[1,3],[2,4],[5],[6]]=>[6,5,2,4,1,3]=>[2,1,6,5,4,3]=>[2,1,5,4,3] [[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[1,3,6,5,4,2]=>[1,3,5,4,2] [[1,6],[2],[3],[4],[5]]=>[5,4,3,2,1,6]=>[5,4,3,2,1,6]=>[5,4,3,2,1] [[1,5],[2],[3],[4],[6]]=>[6,4,3,2,1,5]=>[4,6,3,2,1,5]=>[4,3,2,1,5] [[1,4],[2],[3],[5],[6]]=>[6,5,3,2,1,4]=>[3,6,5,2,1,4]=>[3,5,2,1,4] [[1,3],[2],[4],[5],[6]]=>[6,5,4,2,1,3]=>[2,6,5,4,1,3]=>[2,5,4,1,3] [[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[1,6,5,4,3,2]=>[1,5,4,3,2] [[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[5,4,3,2,1] [[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] [[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] [[1,2,4,5,6,7],[3]]=>[3,1,2,4,5,6,7]=>[1,3,2,4,5,6,7]=>[1,3,2,4,5,6] [[1,2,3,5,6,7],[4]]=>[4,1,2,3,5,6,7]=>[1,2,4,3,5,6,7]=>[1,2,4,3,5,6] [[1,2,3,4,6,7],[5]]=>[5,1,2,3,4,6,7]=>[1,2,3,5,4,6,7]=>[1,2,3,5,4,6] [[1,2,3,4,5,7],[6]]=>[6,1,2,3,4,5,7]=>[1,2,3,4,6,5,7]=>[1,2,3,4,6,5] [[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[1,2,3,4,5,7,6]=>[1,2,3,4,5,6] [[1,3,5,6,7],[2,4]]=>[2,4,1,3,5,6,7]=>[2,1,4,3,5,6,7]=>[2,1,4,3,5,6] [[1,2,5,6,7],[3,4]]=>[3,4,1,2,5,6,7]=>[1,3,4,2,5,6,7]=>[1,3,4,2,5,6] [[1,3,4,6,7],[2,5]]=>[2,5,1,3,4,6,7]=>[2,1,3,5,4,6,7]=>[2,1,3,5,4,6] [[1,2,4,6,7],[3,5]]=>[3,5,1,2,4,6,7]=>[1,3,2,5,4,6,7]=>[1,3,2,5,4,6] [[1,2,3,6,7],[4,5]]=>[4,5,1,2,3,6,7]=>[1,2,4,5,3,6,7]=>[1,2,4,5,3,6] [[1,3,4,5,7],[2,6]]=>[2,6,1,3,4,5,7]=>[2,1,3,4,6,5,7]=>[2,1,3,4,6,5] [[1,2,4,5,7],[3,6]]=>[3,6,1,2,4,5,7]=>[1,3,2,4,6,5,7]=>[1,3,2,4,6,5] [[1,2,3,5,7],[4,6]]=>[4,6,1,2,3,5,7]=>[1,2,4,3,6,5,7]=>[1,2,4,3,6,5] [[1,2,3,4,7],[5,6]]=>[5,6,1,2,3,4,7]=>[1,2,3,5,6,4,7]=>[1,2,3,5,6,4] [[1,3,4,5,6],[2,7]]=>[2,7,1,3,4,5,6]=>[2,1,3,4,5,7,6]=>[2,1,3,4,5,6] [[1,2,4,5,6],[3,7]]=>[3,7,1,2,4,5,6]=>[1,3,2,4,5,7,6]=>[1,3,2,4,5,6] [[1,2,3,5,6],[4,7]]=>[4,7,1,2,3,5,6]=>[1,2,4,3,5,7,6]=>[1,2,4,3,5,6] [[1,2,3,4,6],[5,7]]=>[5,7,1,2,3,4,6]=>[1,2,3,5,4,7,6]=>[1,2,3,5,4,6] [[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[1,2,3,4,6,7,5]=>[1,2,3,4,6,5] [[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] [[1,3,5,6,7],[2],[4]]=>[4,2,1,3,5,6,7]=>[2,4,1,3,5,6,7]=>[2,4,1,3,5,6] [[1,2,5,6,7],[3],[4]]=>[4,3,1,2,5,6,7]=>[1,4,3,2,5,6,7]=>[1,4,3,2,5,6] [[1,3,4,6,7],[2],[5]]=>[5,2,1,3,4,6,7]=>[2,1,5,3,4,6,7]=>[2,1,5,3,4,6] [[1,2,4,6,7],[3],[5]]=>[5,3,1,2,4,6,7]=>[1,3,5,2,4,6,7]=>[1,3,5,2,4,6] [[1,2,3,6,7],[4],[5]]=>[5,4,1,2,3,6,7]=>[1,2,5,4,3,6,7]=>[1,2,5,4,3,6] [[1,3,4,5,7],[2],[6]]=>[6,2,1,3,4,5,7]=>[2,1,3,6,4,5,7]=>[2,1,3,6,4,5] [[1,2,4,5,7],[3],[6]]=>[6,3,1,2,4,5,7]=>[1,3,2,6,4,5,7]=>[1,3,2,6,4,5] [[1,2,3,5,7],[4],[6]]=>[6,4,1,2,3,5,7]=>[1,2,4,6,3,5,7]=>[1,2,4,6,3,5] [[1,2,3,4,7],[5],[6]]=>[6,5,1,2,3,4,7]=>[1,2,3,6,5,4,7]=>[1,2,3,6,5,4] [[1,3,4,5,6],[2],[7]]=>[7,2,1,3,4,5,6]=>[2,1,3,4,7,5,6]=>[2,1,3,4,5,6] [[1,2,4,5,6],[3],[7]]=>[7,3,1,2,4,5,6]=>[1,3,2,4,7,5,6]=>[1,3,2,4,5,6] [[1,2,3,5,6],[4],[7]]=>[7,4,1,2,3,5,6]=>[1,2,4,3,7,5,6]=>[1,2,4,3,5,6] [[1,2,3,4,6],[5],[7]]=>[7,5,1,2,3,4,6]=>[1,2,3,5,7,4,6]=>[1,2,3,5,4,6] [[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[1,2,3,4,7,6,5]=>[1,2,3,4,6,5] [[1,3,5,7],[2,4,6]]=>[2,4,6,1,3,5,7]=>[2,1,4,3,6,5,7]=>[2,1,4,3,6,5] [[1,2,5,7],[3,4,6]]=>[3,4,6,1,2,5,7]=>[1,3,4,2,6,5,7]=>[1,3,4,2,6,5] [[1,3,4,7],[2,5,6]]=>[2,5,6,1,3,4,7]=>[2,1,3,5,6,4,7]=>[2,1,3,5,6,4] [[1,2,4,7],[3,5,6]]=>[3,5,6,1,2,4,7]=>[1,3,2,5,6,4,7]=>[1,3,2,5,6,4] [[1,2,3,7],[4,5,6]]=>[4,5,6,1,2,3,7]=>[1,2,4,5,6,3,7]=>[1,2,4,5,6,3] [[1,3,5,6],[2,4,7]]=>[2,4,7,1,3,5,6]=>[2,1,4,3,5,7,6]=>[2,1,4,3,5,6] [[1,2,5,6],[3,4,7]]=>[3,4,7,1,2,5,6]=>[1,3,4,2,5,7,6]=>[1,3,4,2,5,6] [[1,3,4,6],[2,5,7]]=>[2,5,7,1,3,4,6]=>[2,1,3,5,4,7,6]=>[2,1,3,5,4,6] [[1,2,4,6],[3,5,7]]=>[3,5,7,1,2,4,6]=>[1,3,2,5,4,7,6]=>[1,3,2,5,4,6] [[1,2,3,6],[4,5,7]]=>[4,5,7,1,2,3,6]=>[1,2,4,5,3,7,6]=>[1,2,4,5,3,6] [[1,3,4,5],[2,6,7]]=>[2,6,7,1,3,4,5]=>[2,1,3,4,6,7,5]=>[2,1,3,4,6,5] [[1,2,4,5],[3,6,7]]=>[3,6,7,1,2,4,5]=>[1,3,2,4,6,7,5]=>[1,3,2,4,6,5] [[1,2,3,5],[4,6,7]]=>[4,6,7,1,2,3,5]=>[1,2,4,3,6,7,5]=>[1,2,4,3,6,5] [[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[1,2,3,5,6,7,4]=>[1,2,3,5,6,4] [[1,4,6,7],[2,5],[3]]=>[3,2,5,1,4,6,7]=>[3,2,1,5,4,6,7]=>[3,2,1,5,4,6] [[1,3,6,7],[2,5],[4]]=>[4,2,5,1,3,6,7]=>[2,4,1,5,3,6,7]=>[2,4,1,5,3,6] [[1,2,6,7],[3,5],[4]]=>[4,3,5,1,2,6,7]=>[1,4,3,5,2,6,7]=>[1,4,3,5,2,6] [[1,3,6,7],[2,4],[5]]=>[5,2,4,1,3,6,7]=>[2,1,5,4,3,6,7]=>[2,1,5,4,3,6] [[1,2,6,7],[3,4],[5]]=>[5,3,4,1,2,6,7]=>[1,3,5,4,2,6,7]=>[1,3,5,4,2,6] [[1,4,5,7],[2,6],[3]]=>[3,2,6,1,4,5,7]=>[3,2,1,4,6,5,7]=>[3,2,1,4,6,5] [[1,3,5,7],[2,6],[4]]=>[4,2,6,1,3,5,7]=>[2,4,1,3,6,5,7]=>[2,4,1,3,6,5] [[1,2,5,7],[3,6],[4]]=>[4,3,6,1,2,5,7]=>[1,4,3,2,6,5,7]=>[1,4,3,2,6,5] [[1,3,4,7],[2,6],[5]]=>[5,2,6,1,3,4,7]=>[2,1,5,3,6,4,7]=>[2,1,5,3,6,4] [[1,2,4,7],[3,6],[5]]=>[5,3,6,1,2,4,7]=>[1,3,5,2,6,4,7]=>[1,3,5,2,6,4] [[1,2,3,7],[4,6],[5]]=>[5,4,6,1,2,3,7]=>[1,2,5,4,6,3,7]=>[1,2,5,4,6,3] [[1,3,5,7],[2,4],[6]]=>[6,2,4,1,3,5,7]=>[2,1,4,6,3,5,7]=>[2,1,4,6,3,5] [[1,2,5,7],[3,4],[6]]=>[6,3,4,1,2,5,7]=>[1,3,4,6,2,5,7]=>[1,3,4,6,2,5] [[1,3,4,7],[2,5],[6]]=>[6,2,5,1,3,4,7]=>[2,1,3,6,5,4,7]=>[2,1,3,6,5,4] [[1,2,4,7],[3,5],[6]]=>[6,3,5,1,2,4,7]=>[1,3,2,6,5,4,7]=>[1,3,2,6,5,4] [[1,2,3,7],[4,5],[6]]=>[6,4,5,1,2,3,7]=>[1,2,4,6,5,3,7]=>[1,2,4,6,5,3] [[1,4,5,6],[2,7],[3]]=>[3,2,7,1,4,5,6]=>[3,2,1,4,5,7,6]=>[3,2,1,4,5,6] [[1,3,5,6],[2,7],[4]]=>[4,2,7,1,3,5,6]=>[2,4,1,3,5,7,6]=>[2,4,1,3,5,6] [[1,2,5,6],[3,7],[4]]=>[4,3,7,1,2,5,6]=>[1,4,3,2,5,7,6]=>[1,4,3,2,5,6] [[1,3,4,6],[2,7],[5]]=>[5,2,7,1,3,4,6]=>[2,1,5,3,4,7,6]=>[2,1,5,3,4,6] [[1,2,4,6],[3,7],[5]]=>[5,3,7,1,2,4,6]=>[1,3,5,2,4,7,6]=>[1,3,5,2,4,6] [[1,2,3,6],[4,7],[5]]=>[5,4,7,1,2,3,6]=>[1,2,5,4,3,7,6]=>[1,2,5,4,3,6] [[1,3,4,5],[2,7],[6]]=>[6,2,7,1,3,4,5]=>[2,1,3,6,4,7,5]=>[2,1,3,6,4,5] [[1,2,4,5],[3,7],[6]]=>[6,3,7,1,2,4,5]=>[1,3,2,6,4,7,5]=>[1,3,2,6,4,5] [[1,2,3,5],[4,7],[6]]=>[6,4,7,1,2,3,5]=>[1,2,4,6,3,7,5]=>[1,2,4,6,3,5] [[1,2,3,4],[5,7],[6]]=>[6,5,7,1,2,3,4]=>[1,2,3,6,5,7,4]=>[1,2,3,6,5,4] [[1,3,5,6],[2,4],[7]]=>[7,2,4,1,3,5,6]=>[2,1,4,3,7,5,6]=>[2,1,4,3,5,6] [[1,2,5,6],[3,4],[7]]=>[7,3,4,1,2,5,6]=>[1,3,4,2,7,5,6]=>[1,3,4,2,5,6] [[1,3,4,6],[2,5],[7]]=>[7,2,5,1,3,4,6]=>[2,1,3,5,7,4,6]=>[2,1,3,5,4,6] [[1,2,4,6],[3,5],[7]]=>[7,3,5,1,2,4,6]=>[1,3,2,5,7,4,6]=>[1,3,2,5,4,6] [[1,2,3,6],[4,5],[7]]=>[7,4,5,1,2,3,6]=>[1,2,4,5,7,3,6]=>[1,2,4,5,3,6] [[1,3,4,5],[2,6],[7]]=>[7,2,6,1,3,4,5]=>[2,1,3,4,7,6,5]=>[2,1,3,4,6,5] [[1,2,4,5],[3,6],[7]]=>[7,3,6,1,2,4,5]=>[1,3,2,4,7,6,5]=>[1,3,2,4,6,5] [[1,2,3,5],[4,6],[7]]=>[7,4,6,1,2,3,5]=>[1,2,4,3,7,6,5]=>[1,2,4,3,6,5] [[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[1,2,3,5,7,6,4]=>[1,2,3,5,6,4] [[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] [[1,4,6,7],[2],[3],[5]]=>[5,3,2,1,4,6,7]=>[3,5,2,1,4,6,7]=>[3,5,2,1,4,6] [[1,3,6,7],[2],[4],[5]]=>[5,4,2,1,3,6,7]=>[2,5,4,1,3,6,7]=>[2,5,4,1,3,6] [[1,2,6,7],[3],[4],[5]]=>[5,4,3,1,2,6,7]=>[1,5,4,3,2,6,7]=>[1,5,4,3,2,6] [[1,4,5,7],[2],[3],[6]]=>[6,3,2,1,4,5,7]=>[3,2,6,1,4,5,7]=>[3,2,6,1,4,5] [[1,3,5,7],[2],[4],[6]]=>[6,4,2,1,3,5,7]=>[2,4,6,1,3,5,7]=>[2,4,6,1,3,5] [[1,2,5,7],[3],[4],[6]]=>[6,4,3,1,2,5,7]=>[1,4,6,3,2,5,7]=>[1,4,6,3,2,5] [[1,3,4,7],[2],[5],[6]]=>[6,5,2,1,3,4,7]=>[2,1,6,5,3,4,7]=>[2,1,6,5,3,4] [[1,2,4,7],[3],[5],[6]]=>[6,5,3,1,2,4,7]=>[1,3,6,5,2,4,7]=>[1,3,6,5,2,4] [[1,2,3,7],[4],[5],[6]]=>[6,5,4,1,2,3,7]=>[1,2,6,5,4,3,7]=>[1,2,6,5,4,3] [[1,4,5,6],[2],[3],[7]]=>[7,3,2,1,4,5,6]=>[3,2,1,7,4,5,6]=>[3,2,1,4,5,6] [[1,3,5,6],[2],[4],[7]]=>[7,4,2,1,3,5,6]=>[2,4,1,7,3,5,6]=>[2,4,1,3,5,6] [[1,2,5,6],[3],[4],[7]]=>[7,4,3,1,2,5,6]=>[1,4,3,7,2,5,6]=>[1,4,3,2,5,6] [[1,3,4,6],[2],[5],[7]]=>[7,5,2,1,3,4,6]=>[2,1,5,7,3,4,6]=>[2,1,5,3,4,6] [[1,2,4,6],[3],[5],[7]]=>[7,5,3,1,2,4,6]=>[1,3,5,7,2,4,6]=>[1,3,5,2,4,6] [[1,2,3,6],[4],[5],[7]]=>[7,5,4,1,2,3,6]=>[1,2,5,7,4,3,6]=>[1,2,5,4,3,6] [[1,3,4,5],[2],[6],[7]]=>[7,6,2,1,3,4,5]=>[2,1,3,7,6,4,5]=>[2,1,3,6,4,5] [[1,2,4,5],[3],[6],[7]]=>[7,6,3,1,2,4,5]=>[1,3,2,7,6,4,5]=>[1,3,2,6,4,5] [[1,2,3,5],[4],[6],[7]]=>[7,6,4,1,2,3,5]=>[1,2,4,7,6,3,5]=>[1,2,4,6,3,5] [[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[1,2,3,7,6,5,4]=>[1,2,3,6,5,4] [[1,4,6],[2,5,7],[3]]=>[3,2,5,7,1,4,6]=>[3,2,1,5,4,7,6]=>[3,2,1,5,4,6] [[1,3,6],[2,5,7],[4]]=>[4,2,5,7,1,3,6]=>[2,4,1,5,3,7,6]=>[2,4,1,5,3,6] [[1,2,6],[3,5,7],[4]]=>[4,3,5,7,1,2,6]=>[1,4,3,5,2,7,6]=>[1,4,3,5,2,6] [[1,3,6],[2,4,7],[5]]=>[5,2,4,7,1,3,6]=>[2,1,5,4,3,7,6]=>[2,1,5,4,3,6] [[1,2,6],[3,4,7],[5]]=>[5,3,4,7,1,2,6]=>[1,3,5,4,2,7,6]=>[1,3,5,4,2,6] [[1,4,5],[2,6,7],[3]]=>[3,2,6,7,1,4,5]=>[3,2,1,4,6,7,5]=>[3,2,1,4,6,5] [[1,3,5],[2,6,7],[4]]=>[4,2,6,7,1,3,5]=>[2,4,1,3,6,7,5]=>[2,4,1,3,6,5] [[1,2,5],[3,6,7],[4]]=>[4,3,6,7,1,2,5]=>[1,4,3,2,6,7,5]=>[1,4,3,2,6,5] [[1,3,4],[2,6,7],[5]]=>[5,2,6,7,1,3,4]=>[2,1,5,3,6,7,4]=>[2,1,5,3,6,4] [[1,2,4],[3,6,7],[5]]=>[5,3,6,7,1,2,4]=>[1,3,5,2,6,7,4]=>[1,3,5,2,6,4] [[1,2,3],[4,6,7],[5]]=>[5,4,6,7,1,2,3]=>[1,2,5,4,6,7,3]=>[1,2,5,4,6,3] [[1,3,5],[2,4,7],[6]]=>[6,2,4,7,1,3,5]=>[2,1,4,6,3,7,5]=>[2,1,4,6,3,5] [[1,2,5],[3,4,7],[6]]=>[6,3,4,7,1,2,5]=>[1,3,4,6,2,7,5]=>[1,3,4,6,2,5] [[1,3,4],[2,5,7],[6]]=>[6,2,5,7,1,3,4]=>[2,1,3,6,5,7,4]=>[2,1,3,6,5,4] [[1,2,4],[3,5,7],[6]]=>[6,3,5,7,1,2,4]=>[1,3,2,6,5,7,4]=>[1,3,2,6,5,4] [[1,2,3],[4,5,7],[6]]=>[6,4,5,7,1,2,3]=>[1,2,4,6,5,7,3]=>[1,2,4,6,5,3] [[1,3,5],[2,4,6],[7]]=>[7,2,4,6,1,3,5]=>[2,1,4,3,7,6,5]=>[2,1,4,3,6,5] [[1,2,5],[3,4,6],[7]]=>[7,3,4,6,1,2,5]=>[1,3,4,2,7,6,5]=>[1,3,4,2,6,5] [[1,3,4],[2,5,6],[7]]=>[7,2,5,6,1,3,4]=>[2,1,3,5,7,6,4]=>[2,1,3,5,6,4] [[1,2,4],[3,5,6],[7]]=>[7,3,5,6,1,2,4]=>[1,3,2,5,7,6,4]=>[1,3,2,5,6,4] [[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[1,2,4,5,7,6,3]=>[1,2,4,5,6,3] [[1,4,7],[2,5],[3,6]]=>[3,6,2,5,1,4,7]=>[3,2,1,6,5,4,7]=>[3,2,1,6,5,4] [[1,3,7],[2,5],[4,6]]=>[4,6,2,5,1,3,7]=>[2,4,1,6,5,3,7]=>[2,4,1,6,5,3] [[1,2,7],[3,5],[4,6]]=>[4,6,3,5,1,2,7]=>[1,4,3,6,5,2,7]=>[1,4,3,6,5,2] [[1,3,7],[2,4],[5,6]]=>[5,6,2,4,1,3,7]=>[2,1,5,6,4,3,7]=>[2,1,5,6,4,3] [[1,2,7],[3,4],[5,6]]=>[5,6,3,4,1,2,7]=>[1,3,5,6,4,2,7]=>[1,3,5,6,4,2] [[1,4,6],[2,5],[3,7]]=>[3,7,2,5,1,4,6]=>[3,2,1,5,7,4,6]=>[3,2,1,5,4,6] [[1,3,6],[2,5],[4,7]]=>[4,7,2,5,1,3,6]=>[2,4,1,5,7,3,6]=>[2,4,1,5,3,6] [[1,2,6],[3,5],[4,7]]=>[4,7,3,5,1,2,6]=>[1,4,3,5,7,2,6]=>[1,4,3,5,2,6] [[1,3,6],[2,4],[5,7]]=>[5,7,2,4,1,3,6]=>[2,1,5,4,7,3,6]=>[2,1,5,4,3,6] [[1,2,6],[3,4],[5,7]]=>[5,7,3,4,1,2,6]=>[1,3,5,4,7,2,6]=>[1,3,5,4,2,6] [[1,4,5],[2,6],[3,7]]=>[3,7,2,6,1,4,5]=>[3,2,1,4,7,6,5]=>[3,2,1,4,6,5] [[1,3,5],[2,6],[4,7]]=>[4,7,2,6,1,3,5]=>[2,4,1,3,7,6,5]=>[2,4,1,3,6,5] [[1,2,5],[3,6],[4,7]]=>[4,7,3,6,1,2,5]=>[1,4,3,2,7,6,5]=>[1,4,3,2,6,5] [[1,3,4],[2,6],[5,7]]=>[5,7,2,6,1,3,4]=>[2,1,5,3,7,6,4]=>[2,1,5,3,6,4] [[1,2,4],[3,6],[5,7]]=>[5,7,3,6,1,2,4]=>[1,3,5,2,7,6,4]=>[1,3,5,2,6,4] [[1,2,3],[4,6],[5,7]]=>[5,7,4,6,1,2,3]=>[1,2,5,4,7,6,3]=>[1,2,5,4,6,3] [[1,3,5],[2,4],[6,7]]=>[6,7,2,4,1,3,5]=>[2,1,4,6,7,3,5]=>[2,1,4,6,3,5] [[1,2,5],[3,4],[6,7]]=>[6,7,3,4,1,2,5]=>[1,3,4,6,7,2,5]=>[1,3,4,6,2,5] [[1,3,4],[2,5],[6,7]]=>[6,7,2,5,1,3,4]=>[2,1,3,6,7,5,4]=>[2,1,3,6,5,4] [[1,2,4],[3,5],[6,7]]=>[6,7,3,5,1,2,4]=>[1,3,2,6,7,5,4]=>[1,3,2,6,5,4] [[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[1,2,4,6,7,5,3]=>[1,2,4,6,5,3] [[1,5,7],[2,6],[3],[4]]=>[4,3,2,6,1,5,7]=>[4,3,2,1,6,5,7]=>[4,3,2,1,6,5] [[1,4,7],[2,6],[3],[5]]=>[5,3,2,6,1,4,7]=>[3,5,2,1,6,4,7]=>[3,5,2,1,6,4] [[1,3,7],[2,6],[4],[5]]=>[5,4,2,6,1,3,7]=>[2,5,4,1,6,3,7]=>[2,5,4,1,6,3] [[1,2,7],[3,6],[4],[5]]=>[5,4,3,6,1,2,7]=>[1,5,4,3,6,2,7]=>[1,5,4,3,6,2] [[1,4,7],[2,5],[3],[6]]=>[6,3,2,5,1,4,7]=>[3,2,6,1,5,4,7]=>[3,2,6,1,5,4] [[1,3,7],[2,5],[4],[6]]=>[6,4,2,5,1,3,7]=>[2,4,6,1,5,3,7]=>[2,4,6,1,5,3] [[1,2,7],[3,5],[4],[6]]=>[6,4,3,5,1,2,7]=>[1,4,6,3,5,2,7]=>[1,4,6,3,5,2] [[1,3,7],[2,4],[5],[6]]=>[6,5,2,4,1,3,7]=>[2,1,6,5,4,3,7]=>[2,1,6,5,4,3] [[1,2,7],[3,4],[5],[6]]=>[6,5,3,4,1,2,7]=>[1,3,6,5,4,2,7]=>[1,3,6,5,4,2] [[1,5,6],[2,7],[3],[4]]=>[4,3,2,7,1,5,6]=>[4,3,2,1,5,7,6]=>[4,3,2,1,5,6] [[1,4,6],[2,7],[3],[5]]=>[5,3,2,7,1,4,6]=>[3,5,2,1,4,7,6]=>[3,5,2,1,4,6] [[1,3,6],[2,7],[4],[5]]=>[5,4,2,7,1,3,6]=>[2,5,4,1,3,7,6]=>[2,5,4,1,3,6] [[1,2,6],[3,7],[4],[5]]=>[5,4,3,7,1,2,6]=>[1,5,4,3,2,7,6]=>[1,5,4,3,2,6] [[1,4,5],[2,7],[3],[6]]=>[6,3,2,7,1,4,5]=>[3,2,6,1,4,7,5]=>[3,2,6,1,4,5] [[1,3,5],[2,7],[4],[6]]=>[6,4,2,7,1,3,5]=>[2,4,6,1,3,7,5]=>[2,4,6,1,3,5] [[1,2,5],[3,7],[4],[6]]=>[6,4,3,7,1,2,5]=>[1,4,6,3,2,7,5]=>[1,4,6,3,2,5] [[1,3,4],[2,7],[5],[6]]=>[6,5,2,7,1,3,4]=>[2,1,6,5,3,7,4]=>[2,1,6,5,3,4] [[1,2,4],[3,7],[5],[6]]=>[6,5,3,7,1,2,4]=>[1,3,6,5,2,7,4]=>[1,3,6,5,2,4] [[1,2,3],[4,7],[5],[6]]=>[6,5,4,7,1,2,3]=>[1,2,6,5,4,7,3]=>[1,2,6,5,4,3] [[1,4,6],[2,5],[3],[7]]=>[7,3,2,5,1,4,6]=>[3,2,1,7,5,4,6]=>[3,2,1,5,4,6] [[1,3,6],[2,5],[4],[7]]=>[7,4,2,5,1,3,6]=>[2,4,1,7,5,3,6]=>[2,4,1,5,3,6] [[1,2,6],[3,5],[4],[7]]=>[7,4,3,5,1,2,6]=>[1,4,3,7,5,2,6]=>[1,4,3,5,2,6] [[1,3,6],[2,4],[5],[7]]=>[7,5,2,4,1,3,6]=>[2,1,5,7,4,3,6]=>[2,1,5,4,3,6] [[1,2,6],[3,4],[5],[7]]=>[7,5,3,4,1,2,6]=>[1,3,5,7,4,2,6]=>[1,3,5,4,2,6] [[1,4,5],[2,6],[3],[7]]=>[7,3,2,6,1,4,5]=>[3,2,1,7,4,6,5]=>[3,2,1,4,6,5] [[1,3,5],[2,6],[4],[7]]=>[7,4,2,6,1,3,5]=>[2,4,1,7,3,6,5]=>[2,4,1,3,6,5] [[1,2,5],[3,6],[4],[7]]=>[7,4,3,6,1,2,5]=>[1,4,3,7,2,6,5]=>[1,4,3,2,6,5] [[1,3,4],[2,6],[5],[7]]=>[7,5,2,6,1,3,4]=>[2,1,5,7,3,6,4]=>[2,1,5,3,6,4] [[1,2,4],[3,6],[5],[7]]=>[7,5,3,6,1,2,4]=>[1,3,5,7,2,6,4]=>[1,3,5,2,6,4] [[1,2,3],[4,6],[5],[7]]=>[7,5,4,6,1,2,3]=>[1,2,5,7,4,6,3]=>[1,2,5,4,6,3] [[1,3,5],[2,4],[6],[7]]=>[7,6,2,4,1,3,5]=>[2,1,4,7,6,3,5]=>[2,1,4,6,3,5] [[1,2,5],[3,4],[6],[7]]=>[7,6,3,4,1,2,5]=>[1,3,4,7,6,2,5]=>[1,3,4,6,2,5] [[1,3,4],[2,5],[6],[7]]=>[7,6,2,5,1,3,4]=>[2,1,3,7,6,5,4]=>[2,1,3,6,5,4] [[1,2,4],[3,5],[6],[7]]=>[7,6,3,5,1,2,4]=>[1,3,2,7,6,5,4]=>[1,3,2,6,5,4] [[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[1,2,4,7,6,5,3]=>[1,2,4,6,5,3] [[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] [[1,5,7],[2],[3],[4],[6]]=>[6,4,3,2,1,5,7]=>[4,6,3,2,1,5,7]=>[4,6,3,2,1,5] [[1,4,7],[2],[3],[5],[6]]=>[6,5,3,2,1,4,7]=>[3,6,5,2,1,4,7]=>[3,6,5,2,1,4] [[1,3,7],[2],[4],[5],[6]]=>[6,5,4,2,1,3,7]=>[2,6,5,4,1,3,7]=>[2,6,5,4,1,3] [[1,2,7],[3],[4],[5],[6]]=>[6,5,4,3,1,2,7]=>[1,6,5,4,3,2,7]=>[1,6,5,4,3,2] [[1,5,6],[2],[3],[4],[7]]=>[7,4,3,2,1,5,6]=>[4,3,7,2,1,5,6]=>[4,3,2,1,5,6] [[1,4,6],[2],[3],[5],[7]]=>[7,5,3,2,1,4,6]=>[3,5,7,2,1,4,6]=>[3,5,2,1,4,6] [[1,3,6],[2],[4],[5],[7]]=>[7,5,4,2,1,3,6]=>[2,5,7,4,1,3,6]=>[2,5,4,1,3,6] [[1,2,6],[3],[4],[5],[7]]=>[7,5,4,3,1,2,6]=>[1,5,7,4,3,2,6]=>[1,5,4,3,2,6] [[1,4,5],[2],[3],[6],[7]]=>[7,6,3,2,1,4,5]=>[3,2,7,6,1,4,5]=>[3,2,6,1,4,5] [[1,3,5],[2],[4],[6],[7]]=>[7,6,4,2,1,3,5]=>[2,4,7,6,1,3,5]=>[2,4,6,1,3,5] [[1,2,5],[3],[4],[6],[7]]=>[7,6,4,3,1,2,5]=>[1,4,7,6,3,2,5]=>[1,4,6,3,2,5] [[1,3,4],[2],[5],[6],[7]]=>[7,6,5,2,1,3,4]=>[2,1,7,6,5,3,4]=>[2,1,6,5,3,4] [[1,2,4],[3],[5],[6],[7]]=>[7,6,5,3,1,2,4]=>[1,3,7,6,5,2,4]=>[1,3,6,5,2,4] [[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[1,2,7,6,5,4,3]=>[1,2,6,5,4,3] [[1,5],[2,6],[3,7],[4]]=>[4,3,7,2,6,1,5]=>[4,3,2,1,7,6,5]=>[4,3,2,1,6,5] [[1,4],[2,6],[3,7],[5]]=>[5,3,7,2,6,1,4]=>[3,5,2,1,7,6,4]=>[3,5,2,1,6,4] [[1,3],[2,6],[4,7],[5]]=>[5,4,7,2,6,1,3]=>[2,5,4,1,7,6,3]=>[2,5,4,1,6,3] [[1,2],[3,6],[4,7],[5]]=>[5,4,7,3,6,1,2]=>[1,5,4,3,7,6,2]=>[1,5,4,3,6,2] [[1,4],[2,5],[3,7],[6]]=>[6,3,7,2,5,1,4]=>[3,2,6,1,7,5,4]=>[3,2,6,1,5,4] [[1,3],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3]=>[2,4,6,1,7,5,3]=>[2,4,6,1,5,3] [[1,2],[3,5],[4,7],[6]]=>[6,4,7,3,5,1,2]=>[1,4,6,3,7,5,2]=>[1,4,6,3,5,2] [[1,3],[2,4],[5,7],[6]]=>[6,5,7,2,4,1,3]=>[2,1,6,5,7,4,3]=>[2,1,6,5,4,3] [[1,2],[3,4],[5,7],[6]]=>[6,5,7,3,4,1,2]=>[1,3,6,5,7,4,2]=>[1,3,6,5,4,2] [[1,4],[2,5],[3,6],[7]]=>[7,3,6,2,5,1,4]=>[3,2,1,7,6,5,4]=>[3,2,1,6,5,4] [[1,3],[2,5],[4,6],[7]]=>[7,4,6,2,5,1,3]=>[2,4,1,7,6,5,3]=>[2,4,1,6,5,3] [[1,2],[3,5],[4,6],[7]]=>[7,4,6,3,5,1,2]=>[1,4,3,7,6,5,2]=>[1,4,3,6,5,2] [[1,3],[2,4],[5,6],[7]]=>[7,5,6,2,4,1,3]=>[2,1,5,7,6,4,3]=>[2,1,5,6,4,3] [[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[1,3,5,7,6,4,2]=>[1,3,5,6,4,2] [[1,6],[2,7],[3],[4],[5]]=>[5,4,3,2,7,1,6]=>[5,4,3,2,1,7,6]=>[5,4,3,2,1,6] [[1,5],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5]=>[4,6,3,2,1,7,5]=>[4,6,3,2,1,5] [[1,4],[2,7],[3],[5],[6]]=>[6,5,3,2,7,1,4]=>[3,6,5,2,1,7,4]=>[3,6,5,2,1,4] [[1,3],[2,7],[4],[5],[6]]=>[6,5,4,2,7,1,3]=>[2,6,5,4,1,7,3]=>[2,6,5,4,1,3] [[1,2],[3,7],[4],[5],[6]]=>[6,5,4,3,7,1,2]=>[1,6,5,4,3,7,2]=>[1,6,5,4,3,2] [[1,5],[2,6],[3],[4],[7]]=>[7,4,3,2,6,1,5]=>[4,3,7,2,1,6,5]=>[4,3,2,1,6,5] [[1,4],[2,6],[3],[5],[7]]=>[7,5,3,2,6,1,4]=>[3,5,7,2,1,6,4]=>[3,5,2,1,6,4] [[1,3],[2,6],[4],[5],[7]]=>[7,5,4,2,6,1,3]=>[2,5,7,4,1,6,3]=>[2,5,4,1,6,3] [[1,2],[3,6],[4],[5],[7]]=>[7,5,4,3,6,1,2]=>[1,5,7,4,3,6,2]=>[1,5,4,3,6,2] [[1,4],[2,5],[3],[6],[7]]=>[7,6,3,2,5,1,4]=>[3,2,7,6,1,5,4]=>[3,2,6,1,5,4] [[1,3],[2,5],[4],[6],[7]]=>[7,6,4,2,5,1,3]=>[2,4,7,6,1,5,3]=>[2,4,6,1,5,3] [[1,2],[3,5],[4],[6],[7]]=>[7,6,4,3,5,1,2]=>[1,4,7,6,3,5,2]=>[1,4,6,3,5,2] [[1,3],[2,4],[5],[6],[7]]=>[7,6,5,2,4,1,3]=>[2,1,7,6,5,4,3]=>[2,1,6,5,4,3] [[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[1,3,7,6,5,4,2]=>[1,3,6,5,4,2] [[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] [[1,6],[2],[3],[4],[5],[7]]=>[7,5,4,3,2,1,6]=>[5,7,4,3,2,1,6]=>[5,4,3,2,1,6] [[1,5],[2],[3],[4],[6],[7]]=>[7,6,4,3,2,1,5]=>[4,7,6,3,2,1,5]=>[4,6,3,2,1,5] [[1,4],[2],[3],[5],[6],[7]]=>[7,6,5,3,2,1,4]=>[3,7,6,5,2,1,4]=>[3,6,5,2,1,4] [[1,3],[2],[4],[5],[6],[7]]=>[7,6,5,4,2,1,3]=>[2,7,6,5,4,1,3]=>[2,6,5,4,1,3] [[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[1,7,6,5,4,3,2]=>[1,6,5,4,3,2] [[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[7,6,5,4,3,2,1]=>[6,5,4,3,2,1] [[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] [[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] [[1,2,4,5,6,7,8],[3]]=>[3,1,2,4,5,6,7,8]=>[1,3,2,4,5,6,7,8]=>[1,3,2,4,5,6,7] [[1,2,3,4,6,7,8],[5]]=>[5,1,2,3,4,6,7,8]=>[1,2,3,5,4,6,7,8]=>[1,2,3,5,4,6,7] [[1,2,3,4,5,6,8],[7]]=>[7,1,2,3,4,5,6,8]=>[1,2,3,4,5,7,6,8]=>[1,2,3,4,5,7,6] [[1,2,3,4,5,6,7],[8]]=>[8,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,8,7]=>[1,2,3,4,5,6,7] [[1,2,4,6,7,8],[3,5]]=>[3,5,1,2,4,6,7,8]=>[1,3,2,5,4,6,7,8]=>[1,3,2,5,4,6,7] [[1,2,3,5,7,8],[4,6]]=>[4,6,1,2,3,5,7,8]=>[1,2,4,3,6,5,7,8]=>[1,2,4,3,6,5,7] [[1,2,4,5,6,8],[3,7]]=>[3,7,1,2,4,5,6,8]=>[1,3,2,4,5,7,6,8]=>[1,3,2,4,5,7,6] [[1,2,3,4,6,8],[5,7]]=>[5,7,1,2,3,4,6,8]=>[1,2,3,5,4,7,6,8]=>[1,2,3,5,4,7,6] [[1,3,4,5,6,7],[2,8]]=>[2,8,1,3,4,5,6,7]=>[2,1,3,4,5,6,8,7]=>[2,1,3,4,5,6,7] [[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>[1,2,3,4,5,7,8,6]=>[1,2,3,4,5,7,6] [[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] [[1,2,4,5,7,8],[3],[6]]=>[6,3,1,2,4,5,7,8]=>[1,3,2,6,4,5,7,8]=>[1,3,2,6,4,5,7] [[1,2,4,5,6,7],[3],[8]]=>[8,3,1,2,4,5,6,7]=>[1,3,2,4,5,8,6,7]=>[1,3,2,4,5,6,7] [[1,2,3,4,6,7],[5],[8]]=>[8,5,1,2,3,4,6,7]=>[1,2,3,5,4,8,6,7]=>[1,2,3,5,4,6,7] [[1,2,3,4,5,6],[7],[8]]=>[8,7,1,2,3,4,5,6]=>[1,2,3,4,5,8,7,6]=>[1,2,3,4,5,7,6] [[1,2,4,6,8],[3,5,7]]=>[3,5,7,1,2,4,6,8]=>[1,3,2,5,4,7,6,8]=>[1,3,2,5,4,7,6] [[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>[1,2,3,4,6,7,8,5]=>[1,2,3,4,6,7,5] [[1,3,6,7,8],[2,5],[4]]=>[4,2,5,1,3,6,7,8]=>[2,4,1,5,3,6,7,8]=>[2,4,1,5,3,6,7] [[1,2,3,7,8],[4,5],[6]]=>[6,4,5,1,2,3,7,8]=>[1,2,4,6,5,3,7,8]=>[1,2,4,6,5,3,7] [[1,2,4,5,8],[3,7],[6]]=>[6,3,7,1,2,4,5,8]=>[1,3,2,6,4,7,5,8]=>[1,3,2,6,4,7,5] [[1,2,3,4,5],[6,8],[7]]=>[7,6,8,1,2,3,4,5]=>[1,2,3,4,7,6,8,5]=>[1,2,3,4,7,6,5] [[1,2,4,6,7],[3,5],[8]]=>[8,3,5,1,2,4,6,7]=>[1,3,2,5,4,8,6,7]=>[1,3,2,5,4,6,7] [[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] [[1,4,5,7,8],[2],[3],[6]]=>[6,3,2,1,4,5,7,8]=>[3,2,6,1,4,5,7,8]=>[3,2,6,1,4,5,7] [[1,2,5,6,8],[3],[4],[7]]=>[7,4,3,1,2,5,6,8]=>[1,4,3,7,2,5,6,8]=>[1,4,3,7,2,5,6] [[1,4,5,6,7],[2],[3],[8]]=>[8,3,2,1,4,5,6,7]=>[3,2,1,4,8,5,6,7]=>[3,2,1,4,5,6,7] [[1,3,5,6,7],[2],[4],[8]]=>[8,4,2,1,3,5,6,7]=>[2,4,1,3,8,5,6,7]=>[2,4,1,3,5,6,7] [[1,2,5,6,7],[3],[4],[8]]=>[8,4,3,1,2,5,6,7]=>[1,4,3,2,8,5,6,7]=>[1,4,3,2,5,6,7] [[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[1,2,3,4,8,7,6,5]=>[1,2,3,4,7,6,5] [[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>[2,1,4,3,6,5,8,7]=>[2,1,4,3,6,5,7] [[1,3,4,5],[2,6,7,8]]=>[2,6,7,8,1,3,4,5]=>[2,1,3,4,6,7,8,5]=>[2,1,3,4,6,7,5] [[1,2,4,5],[3,6,7,8]]=>[3,6,7,8,1,2,4,5]=>[1,3,2,4,6,7,8,5]=>[1,3,2,4,6,7,5] [[1,2,3,5],[4,6,7,8]]=>[4,6,7,8,1,2,3,5]=>[1,2,4,3,6,7,8,5]=>[1,2,4,3,6,7,5] [[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[1,2,3,5,6,7,8,4]=>[1,2,3,5,6,7,4] [[1,4,6,8],[2,5,7],[3]]=>[3,2,5,7,1,4,6,8]=>[3,2,1,5,4,7,6,8]=>[3,2,1,5,4,7,6] [[1,3,6,8],[2,5,7],[4]]=>[4,2,5,7,1,3,6,8]=>[2,4,1,5,3,7,6,8]=>[2,4,1,5,3,7,6] [[1,2,4,8],[3,6,7],[5]]=>[5,3,6,7,1,2,4,8]=>[1,3,5,2,6,7,4,8]=>[1,3,5,2,6,7,4] [[1,3,6,7],[2,5,8],[4]]=>[4,2,5,8,1,3,6,7]=>[2,4,1,5,3,6,8,7]=>[2,4,1,5,3,6,7] [[1,3,6,7],[2,4,8],[5]]=>[5,2,4,8,1,3,6,7]=>[2,1,5,4,3,6,8,7]=>[2,1,5,4,3,6,7] [[1,3,5,7],[2,6,8],[4]]=>[4,2,6,8,1,3,5,7]=>[2,4,1,3,6,5,8,7]=>[2,4,1,3,6,5,7] [[1,3,5,7],[2,4,8],[6]]=>[6,2,4,8,1,3,5,7]=>[2,1,4,6,3,5,8,7]=>[2,1,4,6,3,5,7] [[1,2,3,4],[5,7,8],[6]]=>[6,5,7,8,1,2,3,4]=>[1,2,3,6,5,7,8,4]=>[1,2,3,6,5,7,4] [[1,3,5,6],[2,4,8],[7]]=>[7,2,4,8,1,3,5,6]=>[2,1,4,3,7,5,8,6]=>[2,1,4,3,7,5,6] [[1,2,3,6],[4,5,8],[7]]=>[7,4,5,8,1,2,3,6]=>[1,2,4,5,7,3,8,6]=>[1,2,4,5,7,3,6] [[1,3,4,5],[2,6,8],[7]]=>[7,2,6,8,1,3,4,5]=>[2,1,3,4,7,6,8,5]=>[2,1,3,4,7,6,5] [[1,2,4,5],[3,6,8],[7]]=>[7,3,6,8,1,2,4,5]=>[1,3,2,4,7,6,8,5]=>[1,3,2,4,7,6,5] [[1,2,3,5],[4,6,8],[7]]=>[7,4,6,8,1,2,3,5]=>[1,2,4,3,7,6,8,5]=>[1,2,4,3,7,6,5] [[1,2,3,4],[5,6,8],[7]]=>[7,5,6,8,1,2,3,4]=>[1,2,3,5,7,6,8,4]=>[1,2,3,5,7,6,4] [[1,3,5,7],[2,4,6],[8]]=>[8,2,4,6,1,3,5,7]=>[2,1,4,3,6,8,5,7]=>[2,1,4,3,6,5,7] [[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[1,2,3,5,6,8,7,4]=>[1,2,3,5,6,7,4] [[1,4,5,8],[2,6],[3,7]]=>[3,7,2,6,1,4,5,8]=>[3,2,1,4,7,6,5,8]=>[3,2,1,4,7,6,5] [[1,2,5,8],[3,6],[4,7]]=>[4,7,3,6,1,2,5,8]=>[1,4,3,2,7,6,5,8]=>[1,4,3,2,7,6,5] [[1,3,6,7],[2,5],[4,8]]=>[4,8,2,5,1,3,6,7]=>[2,4,1,5,3,8,6,7]=>[2,4,1,5,3,6,7] [[1,3,5,7],[2,6],[4,8]]=>[4,8,2,6,1,3,5,7]=>[2,4,1,3,6,8,5,7]=>[2,4,1,3,6,5,7] [[1,2,4,7],[3,6],[5,8]]=>[5,8,3,6,1,2,4,7]=>[1,3,5,2,6,8,4,7]=>[1,3,5,2,6,4,7] [[1,3,5,7],[2,4],[6,8]]=>[6,8,2,4,1,3,5,7]=>[2,1,4,6,3,8,5,7]=>[2,1,4,6,3,5,7] [[1,2,4,7],[3,5],[6,8]]=>[6,8,3,5,1,2,4,7]=>[1,3,2,6,5,8,4,7]=>[1,3,2,6,5,4,7] [[1,4,5,6],[2,7],[3,8]]=>[3,8,2,7,1,4,5,6]=>[3,2,1,4,5,8,7,6]=>[3,2,1,4,5,7,6] [[1,2,3,4],[5,7],[6,8]]=>[6,8,5,7,1,2,3,4]=>[1,2,3,6,5,8,7,4]=>[1,2,3,6,5,7,4] [[1,3,5,6],[2,4],[7,8]]=>[7,8,2,4,1,3,5,6]=>[2,1,4,3,7,8,5,6]=>[2,1,4,3,7,5,6] [[1,3,4,5],[2,6],[7,8]]=>[7,8,2,6,1,3,4,5]=>[2,1,3,4,7,8,6,5]=>[2,1,3,4,7,6,5] [[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[1,2,3,5,7,8,6,4]=>[1,2,3,5,7,6,4] [[1,5,7,8],[2,6],[3],[4]]=>[4,3,2,6,1,5,7,8]=>[4,3,2,1,6,5,7,8]=>[4,3,2,1,6,5,7] [[1,4,7,8],[2,6],[3],[5]]=>[5,3,2,6,1,4,7,8]=>[3,5,2,1,6,4,7,8]=>[3,5,2,1,6,4,7] [[1,4,7,8],[2,5],[3],[6]]=>[6,3,2,5,1,4,7,8]=>[3,2,6,1,5,4,7,8]=>[3,2,6,1,5,4,7] [[1,5,6,8],[2,7],[3],[4]]=>[4,3,2,7,1,5,6,8]=>[4,3,2,1,5,7,6,8]=>[4,3,2,1,5,7,6] [[1,4,6,8],[2,7],[3],[5]]=>[5,3,2,7,1,4,6,8]=>[3,5,2,1,4,7,6,8]=>[3,5,2,1,4,7,6] [[1,4,5,8],[2,7],[3],[6]]=>[6,3,2,7,1,4,5,8]=>[3,2,6,1,4,7,5,8]=>[3,2,6,1,4,7,5] [[1,5,6,7],[2,8],[3],[4]]=>[4,3,2,8,1,5,6,7]=>[4,3,2,1,5,6,8,7]=>[4,3,2,1,5,6,7] [[1,4,6,7],[2,8],[3],[5]]=>[5,3,2,8,1,4,6,7]=>[3,5,2,1,4,6,8,7]=>[3,5,2,1,4,6,7] [[1,3,5,7],[2,8],[4],[6]]=>[6,4,2,8,1,3,5,7]=>[2,4,6,1,3,5,8,7]=>[2,4,6,1,3,5,7] [[1,3,4,7],[2,8],[5],[6]]=>[6,5,2,8,1,3,4,7]=>[2,1,6,5,3,4,8,7]=>[2,1,6,5,3,4,7] [[1,2,3,4],[5,8],[6],[7]]=>[7,6,5,8,1,2,3,4]=>[1,2,3,7,6,5,8,4]=>[1,2,3,7,6,5,4] [[1,4,6,7],[2,5],[3],[8]]=>[8,3,2,5,1,4,6,7]=>[3,2,1,5,8,4,6,7]=>[3,2,1,5,4,6,7] [[1,3,5,7],[2,6],[4],[8]]=>[8,4,2,6,1,3,5,7]=>[2,4,1,3,8,6,5,7]=>[2,4,1,3,6,5,7] [[1,3,5,7],[2,4],[6],[8]]=>[8,6,2,4,1,3,5,7]=>[2,1,4,6,8,3,5,7]=>[2,1,4,6,3,5,7] [[1,3,4,6],[2,7],[5],[8]]=>[8,5,2,7,1,3,4,6]=>[2,1,5,3,8,4,7,6]=>[2,1,5,3,4,7,6] [[1,2,3,4],[5,7],[6],[8]]=>[8,6,5,7,1,2,3,4]=>[1,2,3,6,8,5,7,4]=>[1,2,3,6,5,7,4] [[1,3,5,6],[2,4],[7],[8]]=>[8,7,2,4,1,3,5,6]=>[2,1,4,3,8,7,5,6]=>[2,1,4,3,7,5,6] [[1,3,4,6],[2,5],[7],[8]]=>[8,7,2,5,1,3,4,6]=>[2,1,3,5,8,7,4,6]=>[2,1,3,5,7,4,6] [[1,2,4,6],[3,5],[7],[8]]=>[8,7,3,5,1,2,4,6]=>[1,3,2,5,8,7,4,6]=>[1,3,2,5,7,4,6] [[1,3,4,5],[2,6],[7],[8]]=>[8,7,2,6,1,3,4,5]=>[2,1,3,4,8,7,6,5]=>[2,1,3,4,7,6,5] [[1,2,4,5],[3,6],[7],[8]]=>[8,7,3,6,1,2,4,5]=>[1,3,2,4,8,7,6,5]=>[1,3,2,4,7,6,5] [[1,2,3,5],[4,6],[7],[8]]=>[8,7,4,6,1,2,3,5]=>[1,2,4,3,8,7,6,5]=>[1,2,4,3,7,6,5] [[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[1,2,3,5,8,7,6,4]=>[1,2,3,5,7,6,4] [[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] [[1,5,6,8],[2],[3],[4],[7]]=>[7,4,3,2,1,5,6,8]=>[4,3,7,2,1,5,6,8]=>[4,3,7,2,1,5,6] [[1,3,4,8],[2],[5],[6],[7]]=>[7,6,5,2,1,3,4,8]=>[2,1,7,6,5,3,4,8]=>[2,1,7,6,5,3,4] [[1,5,6,7],[2],[3],[4],[8]]=>[8,4,3,2,1,5,6,7]=>[4,3,2,8,1,5,6,7]=>[4,3,2,1,5,6,7] [[1,3,5,7],[2],[4],[6],[8]]=>[8,6,4,2,1,3,5,7]=>[2,4,6,8,1,3,5,7]=>[2,4,6,1,3,5,7] [[1,4,5,6],[2],[3],[7],[8]]=>[8,7,3,2,1,4,5,6]=>[3,2,1,8,7,4,5,6]=>[3,2,1,7,4,5,6] [[1,3,4,6],[2],[5],[7],[8]]=>[8,7,5,2,1,3,4,6]=>[2,1,5,8,7,3,4,6]=>[2,1,5,7,3,4,6] [[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[1,2,3,8,7,6,5,4]=>[1,2,3,7,6,5,4] [[1,3,7],[2,4,8],[5,6]]=>[5,6,2,4,8,1,3,7]=>[2,1,5,6,4,3,8,7]=>[2,1,5,6,4,3,7] [[1,3,6],[2,4,8],[5,7]]=>[5,7,2,4,8,1,3,6]=>[2,1,5,4,7,3,8,6]=>[2,1,5,4,7,3,6] [[1,4,5],[2,6,8],[3,7]]=>[3,7,2,6,8,1,4,5]=>[3,2,1,4,7,6,8,5]=>[3,2,1,4,7,6,5] [[1,3,5],[2,6,8],[4,7]]=>[4,7,2,6,8,1,3,5]=>[2,4,1,3,7,6,8,5]=>[2,4,1,3,7,6,5] [[1,2,5],[3,6,8],[4,7]]=>[4,7,3,6,8,1,2,5]=>[1,4,3,2,7,6,8,5]=>[1,4,3,2,7,6,5] [[1,2,4],[3,6,8],[5,7]]=>[5,7,3,6,8,1,2,4]=>[1,3,5,2,7,6,8,4]=>[1,3,5,2,7,6,4] [[1,2,5],[3,4,8],[6,7]]=>[6,7,3,4,8,1,2,5]=>[1,3,4,6,7,2,8,5]=>[1,3,4,6,7,2,5] [[1,3,6],[2,5,7],[4,8]]=>[4,8,2,5,7,1,3,6]=>[2,4,1,5,3,8,7,6]=>[2,4,1,5,3,7,6] [[1,3,5],[2,6,7],[4,8]]=>[4,8,2,6,7,1,3,5]=>[2,4,1,3,6,8,7,5]=>[2,4,1,3,6,7,5] [[1,3,4],[2,6,7],[5,8]]=>[5,8,2,6,7,1,3,4]=>[2,1,5,3,6,8,7,4]=>[2,1,5,3,6,7,4] [[1,3,5],[2,4,7],[6,8]]=>[6,8,2,4,7,1,3,5]=>[2,1,4,6,3,8,7,5]=>[2,1,4,6,3,7,5] [[1,3,5],[2,4,6],[7,8]]=>[7,8,2,4,6,1,3,5]=>[2,1,4,3,7,8,6,5]=>[2,1,4,3,7,6,5] [[1,5,7],[2,6,8],[3],[4]]=>[4,3,2,6,8,1,5,7]=>[4,3,2,1,6,5,8,7]=>[4,3,2,1,6,5,7] [[1,3,7],[2,6,8],[4],[5]]=>[5,4,2,6,8,1,3,7]=>[2,5,4,1,6,3,8,7]=>[2,5,4,1,6,3,7] [[1,4,7],[2,5,8],[3],[6]]=>[6,3,2,5,8,1,4,7]=>[3,2,6,1,5,4,8,7]=>[3,2,6,1,5,4,7] [[1,3,7],[2,5,8],[4],[6]]=>[6,4,2,5,8,1,3,7]=>[2,4,6,1,5,3,8,7]=>[2,4,6,1,5,3,7] [[1,3,7],[2,4,8],[5],[6]]=>[6,5,2,4,8,1,3,7]=>[2,1,6,5,4,3,8,7]=>[2,1,6,5,4,3,7] [[1,5,6],[2,7,8],[3],[4]]=>[4,3,2,7,8,1,5,6]=>[4,3,2,1,5,7,8,6]=>[4,3,2,1,5,7,6] [[1,3,6],[2,7,8],[4],[5]]=>[5,4,2,7,8,1,3,6]=>[2,5,4,1,3,7,8,6]=>[2,5,4,1,3,7,6] [[1,4,5],[2,7,8],[3],[6]]=>[6,3,2,7,8,1,4,5]=>[3,2,6,1,4,7,8,5]=>[3,2,6,1,4,7,5] [[1,2,3],[4,7,8],[5],[6]]=>[6,5,4,7,8,1,2,3]=>[1,2,6,5,4,7,8,3]=>[1,2,6,5,4,7,3] [[1,3,6],[2,5,8],[4],[7]]=>[7,4,2,5,8,1,3,6]=>[2,4,1,7,5,3,8,6]=>[2,4,1,7,5,3,6] [[1,3,4],[2,6,8],[5],[7]]=>[7,5,2,6,8,1,3,4]=>[2,1,5,7,3,6,8,4]=>[2,1,5,7,3,6,4] [[1,3,6],[2,5,7],[4],[8]]=>[8,4,2,5,7,1,3,6]=>[2,4,1,5,8,3,7,6]=>[2,4,1,5,3,7,6] [[1,3,6],[2,4,7],[5],[8]]=>[8,5,2,4,7,1,3,6]=>[2,1,5,4,8,3,7,6]=>[2,1,5,4,3,7,6] [[1,3,4],[2,6,7],[5],[8]]=>[8,5,2,6,7,1,3,4]=>[2,1,5,3,8,6,7,4]=>[2,1,5,3,6,7,4] [[1,3,5],[2,4,7],[6],[8]]=>[8,6,2,4,7,1,3,5]=>[2,1,4,6,8,3,7,5]=>[2,1,4,6,3,7,5] [[1,3,5],[2,4,6],[7],[8]]=>[8,7,2,4,6,1,3,5]=>[2,1,4,3,8,7,6,5]=>[2,1,4,3,7,6,5] [[1,2,5],[3,4,6],[7],[8]]=>[8,7,3,4,6,1,2,5]=>[1,3,4,2,8,7,6,5]=>[1,3,4,2,7,6,5] [[1,5,8],[2,6],[3,7],[4]]=>[4,3,7,2,6,1,5,8]=>[4,3,2,1,7,6,5,8]=>[4,3,2,1,7,6,5] [[1,3,8],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3,8]=>[2,4,6,1,7,5,3,8]=>[2,4,6,1,7,5,3] [[1,5,7],[2,6],[3,8],[4]]=>[4,3,8,2,6,1,5,7]=>[4,3,2,1,6,8,5,7]=>[4,3,2,1,6,5,7] [[1,3,7],[2,5],[4,8],[6]]=>[6,4,8,2,5,1,3,7]=>[2,4,6,1,5,8,3,7]=>[2,4,6,1,5,3,7] [[1,5,6],[2,7],[3,8],[4]]=>[4,3,8,2,7,1,5,6]=>[4,3,2,1,5,8,7,6]=>[4,3,2,1,5,7,6] [[1,3,6],[2,7],[4,8],[5]]=>[5,4,8,2,7,1,3,6]=>[2,5,4,1,3,8,7,6]=>[2,5,4,1,3,7,6] [[1,3,5],[2,7],[4,8],[6]]=>[6,4,8,2,7,1,3,5]=>[2,4,6,1,3,8,7,5]=>[2,4,6,1,3,7,5] [[1,2,6],[3,5],[4,8],[7]]=>[7,4,8,3,5,1,2,6]=>[1,4,3,7,5,8,2,6]=>[1,4,3,7,5,2,6] [[1,3,6],[2,4],[5,8],[7]]=>[7,5,8,2,4,1,3,6]=>[2,1,5,7,4,8,3,6]=>[2,1,5,7,4,3,6] [[1,3,5],[2,6],[4,8],[7]]=>[7,4,8,2,6,1,3,5]=>[2,4,1,7,3,8,6,5]=>[2,4,1,7,3,6,5] [[1,3,4],[2,6],[5,8],[7]]=>[7,5,8,2,6,1,3,4]=>[2,1,5,7,3,8,6,4]=>[2,1,5,7,3,6,4] [[1,4,7],[2,5],[3,6],[8]]=>[8,3,6,2,5,1,4,7]=>[3,2,1,6,8,5,4,7]=>[3,2,1,6,5,4,7] [[1,3,7],[2,5],[4,6],[8]]=>[8,4,6,2,5,1,3,7]=>[2,4,1,6,8,5,3,7]=>[2,4,1,6,5,3,7] [[1,3,5],[2,6],[4,7],[8]]=>[8,4,7,2,6,1,3,5]=>[2,4,1,3,8,7,6,5]=>[2,4,1,3,7,6,5] [[1,2,5],[3,6],[4,7],[8]]=>[8,4,7,3,6,1,2,5]=>[1,4,3,2,8,7,6,5]=>[1,4,3,2,7,6,5] [[1,3,4],[2,6],[5,7],[8]]=>[8,5,7,2,6,1,3,4]=>[2,1,5,3,8,7,6,4]=>[2,1,5,3,7,6,4] [[1,6,8],[2,7],[3],[4],[5]]=>[5,4,3,2,7,1,6,8]=>[5,4,3,2,1,7,6,8]=>[5,4,3,2,1,7,6] [[1,3,8],[2,6],[4],[5],[7]]=>[7,5,4,2,6,1,3,8]=>[2,5,7,4,1,6,3,8]=>[2,5,7,4,1,6,3] [[1,6,7],[2,8],[3],[4],[5]]=>[5,4,3,2,8,1,6,7]=>[5,4,3,2,1,6,8,7]=>[5,4,3,2,1,6,7] [[1,5,6],[2,8],[3],[4],[7]]=>[7,4,3,2,8,1,5,6]=>[4,3,7,2,1,5,8,6]=>[4,3,7,2,1,5,6] [[1,4,7],[2,6],[3],[5],[8]]=>[8,5,3,2,6,1,4,7]=>[3,5,2,8,1,6,4,7]=>[3,5,2,1,6,4,7] [[1,3,7],[2,5],[4],[6],[8]]=>[8,6,4,2,5,1,3,7]=>[2,4,6,8,1,5,3,7]=>[2,4,6,1,5,3,7] [[1,3,5],[2,7],[4],[6],[8]]=>[8,6,4,2,7,1,3,5]=>[2,4,6,8,1,3,7,5]=>[2,4,6,1,3,7,5] [[1,2,5],[3,4],[6],[7],[8]]=>[8,7,6,3,4,1,2,5]=>[1,3,4,8,7,6,2,5]=>[1,3,4,7,6,2,5] [[1,3,4],[2,5],[6],[7],[8]]=>[8,7,6,2,5,1,3,4]=>[2,1,3,8,7,6,5,4]=>[2,1,3,7,6,5,4] [[1,2,4],[3,5],[6],[7],[8]]=>[8,7,6,3,5,1,2,4]=>[1,3,2,8,7,6,5,4]=>[1,3,2,7,6,5,4] [[1,2,3],[4,5],[6],[7],[8]]=>[8,7,6,4,5,1,2,3]=>[1,2,4,8,7,6,5,3]=>[1,2,4,7,6,5,3] [[1,7,8],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7,8]=>[6,5,4,3,2,1,7,8]=>[6,5,4,3,2,1,7] [[1,2,7],[3],[4],[5],[6],[8]]=>[8,6,5,4,3,1,2,7]=>[1,6,8,5,4,3,2,7]=>[1,6,5,4,3,2,7] [[1,3,5],[2],[4],[6],[7],[8]]=>[8,7,6,4,2,1,3,5]=>[2,4,8,7,6,1,3,5]=>[2,4,7,6,1,3,5] [[1,3,4],[2],[5],[6],[7],[8]]=>[8,7,6,5,2,1,3,4]=>[2,1,8,7,6,5,3,4]=>[2,1,7,6,5,3,4] [[1,5],[2,6],[3,7],[4,8]]=>[4,8,3,7,2,6,1,5]=>[4,3,2,1,8,7,6,5]=>[4,3,2,1,7,6,5] [[1,3],[2,4],[5,7],[6,8]]=>[6,8,5,7,2,4,1,3]=>[2,1,6,5,8,7,4,3]=>[2,1,6,5,7,4,3] [[1,4],[2,5],[3,6],[7,8]]=>[7,8,3,6,2,5,1,4]=>[3,2,1,7,8,6,5,4]=>[3,2,1,7,6,5,4] [[1,3],[2,5],[4,6],[7,8]]=>[7,8,4,6,2,5,1,3]=>[2,4,1,7,8,6,5,3]=>[2,4,1,7,6,5,3] [[1,2],[3,5],[4,6],[7,8]]=>[7,8,4,6,3,5,1,2]=>[1,4,3,7,8,6,5,2]=>[1,4,3,7,6,5,2] [[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[1,3,5,7,8,6,4,2]=>[1,3,5,7,6,4,2] [[1,6],[2,7],[3,8],[4],[5]]=>[5,4,3,8,2,7,1,6]=>[5,4,3,2,1,8,7,6]=>[5,4,3,2,1,7,6] [[1,2],[3,7],[4,8],[5],[6]]=>[6,5,4,8,3,7,1,2]=>[1,6,5,4,3,8,7,2]=>[1,6,5,4,3,7,2] [[1,3],[2,4],[5,8],[6],[7]]=>[7,6,5,8,2,4,1,3]=>[2,1,7,6,5,8,4,3]=>[2,1,7,6,5,4,3] [[1,4],[2,6],[3,7],[5],[8]]=>[8,5,3,7,2,6,1,4]=>[3,5,2,8,1,7,6,4]=>[3,5,2,1,7,6,4] [[1,2],[3,5],[4,7],[6],[8]]=>[8,6,4,7,3,5,1,2]=>[1,4,6,8,3,7,5,2]=>[1,4,6,3,7,5,2] [[1,3],[2,4],[5,7],[6],[8]]=>[8,6,5,7,2,4,1,3]=>[2,1,6,8,5,7,4,3]=>[2,1,6,5,7,4,3] [[1,4],[2,5],[3,6],[7],[8]]=>[8,7,3,6,2,5,1,4]=>[3,2,1,8,7,6,5,4]=>[3,2,1,7,6,5,4] [[1,3],[2,5],[4,6],[7],[8]]=>[8,7,4,6,2,5,1,3]=>[2,4,1,8,7,6,5,3]=>[2,4,1,7,6,5,3] [[1,2],[3,5],[4,6],[7],[8]]=>[8,7,4,6,3,5,1,2]=>[1,4,3,8,7,6,5,2]=>[1,4,3,7,6,5,2] [[1,7],[2,8],[3],[4],[5],[6]]=>[6,5,4,3,2,8,1,7]=>[6,5,4,3,2,1,8,7]=>[6,5,4,3,2,1,7] [[1,6],[2,8],[3],[4],[5],[7]]=>[7,5,4,3,2,8,1,6]=>[5,7,4,3,2,1,8,6]=>[5,7,4,3,2,1,6] [[1,4],[2,8],[3],[5],[6],[7]]=>[7,6,5,3,2,8,1,4]=>[3,7,6,5,2,1,8,4]=>[3,7,6,5,2,1,4] [[1,5],[2,6],[3],[4],[7],[8]]=>[8,7,4,3,2,6,1,5]=>[4,3,8,7,2,1,6,5]=>[4,3,7,2,1,6,5] [[1,3],[2,4],[5],[6],[7],[8]]=>[8,7,6,5,2,4,1,3]=>[2,1,8,7,6,5,4,3]=>[2,1,7,6,5,4,3] [[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] [[1,3],[2],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,2,1,3]=>[2,8,7,6,5,4,1,3]=>[2,7,6,5,4,1,3] [[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[1,8,7,6,5,4,3,2]=>[1,7,6,5,4,3,2] [[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1]=>[7,6,5,4,3,2,1] [[1,3,5,7,9],[2,4,6,8,10]]=>[2,4,6,8,10,1,3,5,7,9]=>[2,1,4,3,6,5,8,7,10,9]=>[2,1,4,3,6,5,8,7,9] [[1,2,3,4,5],[6,7,8,9,10]]=>[6,7,8,9,10,1,2,3,4,5]=>[1,2,3,4,6,7,8,9,10,5]=>[1,2,3,4,6,7,8,9,5] [[1,3,5,7,9,11],[2,4,6,8,10,12]]=>[2,4,6,8,10,12,1,3,5,7,9,11]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[2,1,4,3,6,5,8,7,10,9,11] [[1,2,3,4,5,6,7,8,9]]=>[1,2,3,4,5,6,7,8,9]=>[1,2,3,4,5,6,7,8,9]=>[1,2,3,4,5,6,7,8] [[1,2,3,4,5,6,7,8],[9]]=>[9,1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,9,8]=>[1,2,3,4,5,6,7,8] [[1,2,3,4,5,6,7],[8,9]]=>[8,9,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,8,9,7]=>[1,2,3,4,5,6,8,7] [[1,2,3,4,5,6,7],[8],[9]]=>[9,8,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,9,8,7]=>[1,2,3,4,5,6,8,7] [[1,2,3,4,5,6],[7,8,9]]=>[7,8,9,1,2,3,4,5,6]=>[1,2,3,4,5,7,8,9,6]=>[1,2,3,4,5,7,8,6] [[1,2,3,4,5],[6,7,8,9]]=>[6,7,8,9,1,2,3,4,5]=>[1,2,3,4,6,7,8,9,5]=>[1,2,3,4,6,7,8,5] [[1,2],[3],[4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,4,3,1,2]=>[1,9,8,7,6,5,4,3,2]=>[1,8,7,6,5,4,3,2] [[1],[2],[3],[4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,4,3,2,1]=>[9,8,7,6,5,4,3,2,1]=>[8,7,6,5,4,3,2,1] [[1,2,3,4,5,6,7,8,9,10]]=>[1,2,3,4,5,6,7,8,9,10]=>[1,2,3,4,5,6,7,8,9,10]=>[1,2,3,4,5,6,7,8,9] [[1,2,3,4,5,6,7,8,9],[10]]=>[10,1,2,3,4,5,6,7,8,9]=>[1,2,3,4,5,6,7,8,10,9]=>[1,2,3,4,5,6,7,8,9] [[1,2,3,4,5,6,7,8],[9,10]]=>[9,10,1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,9,10,8]=>[1,2,3,4,5,6,7,9,8] [[1,2,3,4,5,6,7],[8,9,10]]=>[8,9,10,1,2,3,4,5,6,7]=>[1,2,3,4,5,6,8,9,10,7]=>[1,2,3,4,5,6,8,9,7] [[1,2,3,4,5,6],[7,8,9,10]]=>[7,8,9,10,1,2,3,4,5,6]=>[1,2,3,4,5,7,8,9,10,6]=>[1,2,3,4,5,7,8,9,6] [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,4,3,1,2]=>[1,10,9,8,7,6,5,4,3,2]=>[1,9,8,7,6,5,4,3,2] [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,4,3,2,1]=>[10,9,8,7,6,5,4,3,2,1]=>[9,8,7,6,5,4,3,2,1] [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]=>[12,11,10,9,8,7,6,5,4,3,2,1]=>[12,11,10,9,8,7,6,5,4,3,2,1]=>[11,10,9,8,7,6,5,4,3,2,1] [[1,3,4,5,6,7,8,9],[2]]=>[2,1,3,4,5,6,7,8,9]=>[2,1,3,4,5,6,7,8,9]=>[2,1,3,4,5,6,7,8] [[1,4,5,6,7,8,9],[2],[3]]=>[3,2,1,4,5,6,7,8,9]=>[3,2,1,4,5,6,7,8,9]=>[3,2,1,4,5,6,7,8] [[1,5,6,7,8,9],[2],[3],[4]]=>[4,3,2,1,5,6,7,8,9]=>[4,3,2,1,5,6,7,8,9]=>[4,3,2,1,5,6,7,8] [[1,6,7,8,9],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7,8,9]=>[5,4,3,2,1,6,7,8,9]=>[5,4,3,2,1,6,7,8] [[1,7,8,9],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7,8,9]=>[6,5,4,3,2,1,7,8,9]=>[6,5,4,3,2,1,7,8] [[1,8,9],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1,8,9]=>[7,6,5,4,3,2,1,8,9]=>[7,6,5,4,3,2,1,8] [[1,9],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1,9]=>[8,7,6,5,4,3,2,1,9]=>[8,7,6,5,4,3,2,1] [[1,3,4,5,6,7,8,9,10],[2]]=>[2,1,3,4,5,6,7,8,9,10]=>[2,1,3,4,5,6,7,8,9,10]=>[2,1,3,4,5,6,7,8,9] [[1,4,5,6,7,8,9,10],[2],[3]]=>[3,2,1,4,5,6,7,8,9,10]=>[3,2,1,4,5,6,7,8,9,10]=>[3,2,1,4,5,6,7,8,9] [[1,5,6,7,8,9,10],[2],[3],[4]]=>[4,3,2,1,5,6,7,8,9,10]=>[4,3,2,1,5,6,7,8,9,10]=>[4,3,2,1,5,6,7,8,9] [[1,6,7,8,9,10],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7,8,9,10]=>[5,4,3,2,1,6,7,8,9,10]=>[5,4,3,2,1,6,7,8,9] [[1,7,8,9,10],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7,8,9,10]=>[6,5,4,3,2,1,7,8,9,10]=>[6,5,4,3,2,1,7,8,9] [[1,8,9,10],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1,8,9,10]=>[7,6,5,4,3,2,1,8,9,10]=>[7,6,5,4,3,2,1,8,9] [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1,9,10]=>[8,7,6,5,4,3,2,1,9,10]=>[8,7,6,5,4,3,2,1,9] [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,4,3,2,1,10]=>[9,8,7,6,5,4,3,2,1,10]=>[9,8,7,6,5,4,3,2,1] [[1,3,5,7,9],[2,4,6,8]]=>[2,4,6,8,1,3,5,7,9]=>[2,1,4,3,6,5,8,7,9]=>[2,1,4,3,6,5,8,7] [[1,9],[2,10],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,10,1,9]=>[8,7,6,5,4,3,2,1,10,9]=>[8,7,6,5,4,3,2,1,9] [[1,7],[2,8],[3,9],[4,10],[5],[6]]=>[6,5,4,10,3,9,2,8,1,7]=>[6,5,4,3,2,1,10,9,8,7]=>[6,5,4,3,2,1,9,8,7] [[1,7,9],[2,8,10],[3],[4],[5],[6]]=>[6,5,4,3,2,8,10,1,7,9]=>[6,5,4,3,2,1,8,7,10,9]=>[6,5,4,3,2,1,8,7,9] [[1,5,9],[2,6,10],[3,7],[4,8]]=>[4,8,3,7,2,6,10,1,5,9]=>[4,3,2,1,8,7,6,5,10,9]=>[4,3,2,1,8,7,6,5,9] [[1,5,7,9],[2,6,8,10],[3],[4]]=>[4,3,2,6,8,10,1,5,7,9]=>[4,3,2,1,6,5,8,7,10,9]=>[4,3,2,1,6,5,8,7,9] [[1,2,4,6,8],[3,5,7,9]]=>[3,5,7,9,1,2,4,6,8]=>[1,3,2,5,4,7,6,9,8]=>[1,3,2,5,4,7,6,8] [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,2,4,1,3]=>[2,1,10,9,8,7,6,5,4,3]=>[2,1,9,8,7,6,5,4,3] [[1,5],[2,6],[3,7],[4,8],[9],[10]]=>[10,9,4,8,3,7,2,6,1,5]=>[4,3,2,1,10,9,8,7,6,5]=>[4,3,2,1,9,8,7,6,5] [[1,3,5],[2,4,6],[7],[8],[9],[10]]=>[10,9,8,7,2,4,6,1,3,5]=>[2,1,4,3,10,9,8,7,6,5]=>[2,1,4,3,9,8,7,6,5] [[1,3,7],[2,4,8],[5,9],[6,10]]=>[6,10,5,9,2,4,8,1,3,7]=>[2,1,6,5,4,3,10,9,8,7]=>[2,1,6,5,4,3,9,8,7] [[1,3,5,7],[2,4,6,8],[9],[10]]=>[10,9,2,4,6,8,1,3,5,7]=>[2,1,4,3,6,5,10,9,8,7]=>[2,1,4,3,6,5,9,8,7] [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]=>[6,12,5,11,4,10,3,9,2,8,1,7]=>[6,5,4,3,2,1,12,11,10,9,8,7]=>[6,5,4,3,2,1,11,10,9,8,7] [[1,2,4,6,8,10],[3,5,7,9]]=>[3,5,7,9,1,2,4,6,8,10]=>[1,3,2,5,4,7,6,9,8,10]=>[1,3,2,5,4,7,6,9,8] [[1,2,3,4,5,6],[7,9],[8]]=>[8,7,9,1,2,3,4,5,6]=>[1,2,3,4,5,8,7,9,6]=>[1,2,3,4,5,8,7,6]
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
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.
  • 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 $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
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
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.