Identifier
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Images
=>
Cc0007;cc-rep-0Cc0010;cc-rep-3
[[1]]=>[1]=>[1]=>[.,.] [[1,2]]=>[1,2]=>[1,2]=>[.,[.,.]] [[1],[2]]=>[2,1]=>[2,1]=>[[.,.],.] [[1,2,3]]=>[1,2,3]=>[1,2,3]=>[.,[.,[.,.]]] [[1,3],[2]]=>[2,1,3]=>[2,1,3]=>[[.,.],[.,.]] [[1,2],[3]]=>[3,1,2]=>[3,1,2]=>[[.,[.,.]],.] [[1],[2],[3]]=>[3,2,1]=>[2,3,1]=>[[.,.],[.,.]] [[1,2,3,4]]=>[1,2,3,4]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]] [[1,3,4],[2]]=>[2,1,3,4]=>[2,1,3,4]=>[[.,.],[.,[.,.]]] [[1,2,4],[3]]=>[3,1,2,4]=>[3,1,2,4]=>[[.,[.,.]],[.,.]] [[1,2,3],[4]]=>[4,1,2,3]=>[4,1,2,3]=>[[.,[.,[.,.]]],.] [[1,3],[2,4]]=>[2,4,1,3]=>[4,2,1,3]=>[[[.,.],[.,.]],.] [[1,2],[3,4]]=>[3,4,1,2]=>[4,1,3,2]=>[[.,[[.,.],.]],.] [[1,4],[2],[3]]=>[3,2,1,4]=>[2,3,1,4]=>[[.,.],[.,[.,.]]] [[1,3],[2],[4]]=>[4,2,1,3]=>[2,4,1,3]=>[[.,.],[[.,.],.]] [[1,2],[3],[4]]=>[4,3,1,2]=>[3,1,4,2]=>[[.,[.,.]],[.,.]] [[1],[2],[3],[4]]=>[4,3,2,1]=>[2,3,4,1]=>[[.,.],[.,[.,.]]] [[1,2,3,4,5]]=>[1,2,3,4,5]=>[1,2,3,4,5]=>[.,[.,[.,[.,[.,.]]]]] [[1,3,4,5],[2]]=>[2,1,3,4,5]=>[2,1,3,4,5]=>[[.,.],[.,[.,[.,.]]]] [[1,2,4,5],[3]]=>[3,1,2,4,5]=>[3,1,2,4,5]=>[[.,[.,.]],[.,[.,.]]] [[1,2,3,5],[4]]=>[4,1,2,3,5]=>[4,1,2,3,5]=>[[.,[.,[.,.]]],[.,.]] [[1,2,3,4],[5]]=>[5,1,2,3,4]=>[5,1,2,3,4]=>[[.,[.,[.,[.,.]]]],.] [[1,3,5],[2,4]]=>[2,4,1,3,5]=>[4,2,1,3,5]=>[[[.,.],[.,.]],[.,.]] [[1,2,5],[3,4]]=>[3,4,1,2,5]=>[4,1,3,2,5]=>[[.,[[.,.],.]],[.,.]] [[1,3,4],[2,5]]=>[2,5,1,3,4]=>[5,2,1,3,4]=>[[[.,.],[.,[.,.]]],.] [[1,2,4],[3,5]]=>[3,5,1,2,4]=>[5,1,3,2,4]=>[[.,[[.,.],[.,.]]],.] [[1,2,3],[4,5]]=>[4,5,1,2,3]=>[5,1,2,4,3]=>[[.,[.,[[.,.],.]]],.] [[1,4,5],[2],[3]]=>[3,2,1,4,5]=>[2,3,1,4,5]=>[[.,.],[.,[.,[.,.]]]] [[1,3,5],[2],[4]]=>[4,2,1,3,5]=>[2,4,1,3,5]=>[[.,.],[[.,.],[.,.]]] [[1,2,5],[3],[4]]=>[4,3,1,2,5]=>[3,1,4,2,5]=>[[.,[.,.]],[.,[.,.]]] [[1,3,4],[2],[5]]=>[5,2,1,3,4]=>[2,5,1,3,4]=>[[.,.],[[.,[.,.]],.]] [[1,2,4],[3],[5]]=>[5,3,1,2,4]=>[3,1,5,2,4]=>[[.,[.,.]],[[.,.],.]] [[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[4,1,2,5,3]=>[[.,[.,[.,.]]],[.,.]] [[1,4],[2,5],[3]]=>[3,2,5,1,4]=>[5,3,2,1,4]=>[[[[.,.],.],[.,.]],.] [[1,3],[2,5],[4]]=>[4,2,5,1,3]=>[5,4,1,2,3]=>[[[.,[.,[.,.]]],.],.] [[1,2],[3,5],[4]]=>[4,3,5,1,2]=>[5,1,4,3,2]=>[[.,[[[.,.],.],.]],.] [[1,3],[2,4],[5]]=>[5,2,4,1,3]=>[4,5,1,2,3]=>[[.,[.,[.,.]]],[.,.]] [[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[4,1,5,3,2]=>[[.,[[.,.],.]],[.,.]] [[1,5],[2],[3],[4]]=>[4,3,2,1,5]=>[2,3,4,1,5]=>[[.,.],[.,[.,[.,.]]]] [[1,4],[2],[3],[5]]=>[5,3,2,1,4]=>[2,3,5,1,4]=>[[.,.],[.,[[.,.],.]]] [[1,3],[2],[4],[5]]=>[5,4,2,1,3]=>[2,4,1,5,3]=>[[.,.],[[.,.],[.,.]]] [[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[3,1,4,5,2]=>[[.,[.,.]],[.,[.,.]]] [[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[2,3,4,5,1]=>[[.,.],[.,[.,[.,.]]]] [[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]=>[.,[.,[.,[.,[.,[.,.]]]]]] [[1,3,4,5,6],[2]]=>[2,1,3,4,5,6]=>[2,1,3,4,5,6]=>[[.,.],[.,[.,[.,[.,.]]]]] [[1,2,4,5,6],[3]]=>[3,1,2,4,5,6]=>[3,1,2,4,5,6]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,2,3,5,6],[4]]=>[4,1,2,3,5,6]=>[4,1,2,3,5,6]=>[[.,[.,[.,.]]],[.,[.,.]]] [[1,2,3,4,6],[5]]=>[5,1,2,3,4,6]=>[5,1,2,3,4,6]=>[[.,[.,[.,[.,.]]]],[.,.]] [[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[6,1,2,3,4,5]=>[[.,[.,[.,[.,[.,.]]]]],.] [[1,3,5,6],[2,4]]=>[2,4,1,3,5,6]=>[4,2,1,3,5,6]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,2,5,6],[3,4]]=>[3,4,1,2,5,6]=>[4,1,3,2,5,6]=>[[.,[[.,.],.]],[.,[.,.]]] [[1,3,4,6],[2,5]]=>[2,5,1,3,4,6]=>[5,2,1,3,4,6]=>[[[.,.],[.,[.,.]]],[.,.]] [[1,2,4,6],[3,5]]=>[3,5,1,2,4,6]=>[5,1,3,2,4,6]=>[[.,[[.,.],[.,.]]],[.,.]] [[1,2,3,6],[4,5]]=>[4,5,1,2,3,6]=>[5,1,2,4,3,6]=>[[.,[.,[[.,.],.]]],[.,.]] [[1,3,4,5],[2,6]]=>[2,6,1,3,4,5]=>[6,2,1,3,4,5]=>[[[.,.],[.,[.,[.,.]]]],.] [[1,2,4,5],[3,6]]=>[3,6,1,2,4,5]=>[6,1,3,2,4,5]=>[[.,[[.,.],[.,[.,.]]]],.] [[1,2,3,5],[4,6]]=>[4,6,1,2,3,5]=>[6,1,2,4,3,5]=>[[.,[.,[[.,.],[.,.]]]],.] [[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[6,1,2,3,5,4]=>[[.,[.,[.,[[.,.],.]]]],.] [[1,4,5,6],[2],[3]]=>[3,2,1,4,5,6]=>[2,3,1,4,5,6]=>[[.,.],[.,[.,[.,[.,.]]]]] [[1,3,5,6],[2],[4]]=>[4,2,1,3,5,6]=>[2,4,1,3,5,6]=>[[.,.],[[.,.],[.,[.,.]]]] [[1,2,5,6],[3],[4]]=>[4,3,1,2,5,6]=>[3,1,4,2,5,6]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,3,4,6],[2],[5]]=>[5,2,1,3,4,6]=>[2,5,1,3,4,6]=>[[.,.],[[.,[.,.]],[.,.]]] [[1,2,4,6],[3],[5]]=>[5,3,1,2,4,6]=>[3,1,5,2,4,6]=>[[.,[.,.]],[[.,.],[.,.]]] [[1,2,3,6],[4],[5]]=>[5,4,1,2,3,6]=>[4,1,2,5,3,6]=>[[.,[.,[.,.]]],[.,[.,.]]] [[1,3,4,5],[2],[6]]=>[6,2,1,3,4,5]=>[2,6,1,3,4,5]=>[[.,.],[[.,[.,[.,.]]],.]] [[1,2,4,5],[3],[6]]=>[6,3,1,2,4,5]=>[3,1,6,2,4,5]=>[[.,[.,.]],[[.,[.,.]],.]] [[1,2,3,5],[4],[6]]=>[6,4,1,2,3,5]=>[4,1,2,6,3,5]=>[[.,[.,[.,.]]],[[.,.],.]] [[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[5,1,2,3,6,4]=>[[.,[.,[.,[.,.]]]],[.,.]] [[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[6,2,1,4,3,5]=>[[[.,.],[[.,.],[.,.]]],.] [[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[6,1,3,4,2,5]=>[[.,[[.,.],[.,[.,.]]]],.] [[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[6,2,1,3,5,4]=>[[[.,.],[.,[[.,.],.]]],.] [[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[6,1,3,2,5,4]=>[[.,[[.,.],[[.,.],.]]],.] [[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]=>[[.,[.,[[.,.],[.,.]]]],.] [[1,4,6],[2,5],[3]]=>[3,2,5,1,4,6]=>[5,3,2,1,4,6]=>[[[[.,.],.],[.,.]],[.,.]] [[1,3,6],[2,5],[4]]=>[4,2,5,1,3,6]=>[5,4,1,2,3,6]=>[[[.,[.,[.,.]]],.],[.,.]] [[1,2,6],[3,5],[4]]=>[4,3,5,1,2,6]=>[5,1,4,3,2,6]=>[[.,[[[.,.],.],.]],[.,.]] [[1,3,6],[2,4],[5]]=>[5,2,4,1,3,6]=>[4,5,1,2,3,6]=>[[.,[.,[.,.]]],[.,[.,.]]] [[1,2,6],[3,4],[5]]=>[5,3,4,1,2,6]=>[4,1,5,3,2,6]=>[[.,[[.,.],.]],[.,[.,.]]] [[1,4,5],[2,6],[3]]=>[3,2,6,1,4,5]=>[6,3,2,1,4,5]=>[[[[.,.],.],[.,[.,.]]],.] [[1,3,5],[2,6],[4]]=>[4,2,6,1,3,5]=>[6,4,1,2,3,5]=>[[[.,[.,[.,.]]],[.,.]],.] [[1,2,5],[3,6],[4]]=>[4,3,6,1,2,5]=>[6,1,4,3,2,5]=>[[.,[[[.,.],.],[.,.]]],.] [[1,3,4],[2,6],[5]]=>[5,2,6,1,3,4]=>[6,5,1,3,2,4]=>[[[.,[[.,.],[.,.]]],.],.] [[1,2,4],[3,6],[5]]=>[5,3,6,1,2,4]=>[6,1,5,2,3,4]=>[[.,[[.,[.,[.,.]]],.]],.] [[1,2,3],[4,6],[5]]=>[5,4,6,1,2,3]=>[6,1,2,5,4,3]=>[[.,[.,[[[.,.],.],.]]],.] [[1,3,5],[2,4],[6]]=>[6,2,4,1,3,5]=>[4,6,1,2,3,5]=>[[.,[.,[.,.]]],[[.,.],.]] [[1,2,5],[3,4],[6]]=>[6,3,4,1,2,5]=>[4,1,6,3,2,5]=>[[.,[[.,.],.]],[[.,.],.]] [[1,3,4],[2,5],[6]]=>[6,2,5,1,3,4]=>[5,6,1,3,2,4]=>[[.,[[.,.],[.,.]]],[.,.]] [[1,2,4],[3,5],[6]]=>[6,3,5,1,2,4]=>[5,1,6,2,3,4]=>[[.,[.,[.,[.,.]]]],[.,.]] [[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[5,1,2,6,4,3]=>[[.,[.,[[.,.],.]]],[.,.]] [[1,5,6],[2],[3],[4]]=>[4,3,2,1,5,6]=>[2,3,4,1,5,6]=>[[.,.],[.,[.,[.,[.,.]]]]] [[1,4,6],[2],[3],[5]]=>[5,3,2,1,4,6]=>[2,3,5,1,4,6]=>[[.,.],[.,[[.,.],[.,.]]]] [[1,3,6],[2],[4],[5]]=>[5,4,2,1,3,6]=>[2,4,1,5,3,6]=>[[.,.],[[.,.],[.,[.,.]]]] [[1,2,6],[3],[4],[5]]=>[5,4,3,1,2,6]=>[3,1,4,5,2,6]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,4,5],[2],[3],[6]]=>[6,3,2,1,4,5]=>[2,3,6,1,4,5]=>[[.,.],[.,[[.,[.,.]],.]]] [[1,3,5],[2],[4],[6]]=>[6,4,2,1,3,5]=>[2,4,1,6,3,5]=>[[.,.],[[.,.],[[.,.],.]]] [[1,2,5],[3],[4],[6]]=>[6,4,3,1,2,5]=>[3,1,4,6,2,5]=>[[.,[.,.]],[.,[[.,.],.]]] [[1,3,4],[2],[5],[6]]=>[6,5,2,1,3,4]=>[2,5,1,3,6,4]=>[[.,.],[[.,[.,.]],[.,.]]] [[1,2,4],[3],[5],[6]]=>[6,5,3,1,2,4]=>[3,1,5,2,6,4]=>[[.,[.,.]],[[.,.],[.,.]]] [[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[4,1,2,5,6,3]=>[[.,[.,[.,.]]],[.,[.,.]]] [[1,4],[2,5],[3,6]]=>[3,6,2,5,1,4]=>[5,6,3,1,2,4]=>[[[.,[.,.]],[.,.]],[.,.]] [[1,3],[2,5],[4,6]]=>[4,6,2,5,1,3]=>[5,6,1,4,2,3]=>[[.,[[.,[.,.]],.]],[.,.]] [[1,2],[3,5],[4,6]]=>[4,6,3,5,1,2]=>[5,1,6,4,3,2]=>[[.,[[[.,.],.],.]],[.,.]] [[1,3],[2,4],[5,6]]=>[5,6,2,4,1,3]=>[4,6,1,2,5,3]=>[[.,[.,[.,.]]],[[.,.],.]] [[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[4,1,6,3,5,2]=>[[.,[[.,.],.]],[[.,.],.]] [[1,5],[2,6],[3],[4]]=>[4,3,2,6,1,5]=>[6,3,4,2,1,5]=>[[[[.,.],.],[.,[.,.]]],.] [[1,4],[2,6],[3],[5]]=>[5,3,2,6,1,4]=>[6,3,5,1,2,4]=>[[[.,[.,.]],[[.,.],.]],.] [[1,3],[2,6],[4],[5]]=>[5,4,2,6,1,3]=>[6,4,1,5,2,3]=>[[[.,[.,[.,.]]],[.,.]],.] [[1,2],[3,6],[4],[5]]=>[5,4,3,6,1,2]=>[6,1,4,5,3,2]=>[[.,[[[.,.],.],[.,.]]],.] [[1,4],[2,5],[3],[6]]=>[6,3,2,5,1,4]=>[5,3,6,1,2,4]=>[[[.,[.,.]],[.,.]],[.,.]] [[1,3],[2,5],[4],[6]]=>[6,4,2,5,1,3]=>[5,4,1,6,2,3]=>[[[.,[.,[.,.]]],.],[.,.]] [[1,2],[3,5],[4],[6]]=>[6,4,3,5,1,2]=>[5,1,4,6,3,2]=>[[.,[[[.,.],.],.]],[.,.]] [[1,3],[2,4],[5],[6]]=>[6,5,2,4,1,3]=>[4,5,1,2,6,3]=>[[.,[.,[.,.]]],[.,[.,.]]] [[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[4,1,5,3,6,2]=>[[.,[[.,.],.]],[.,[.,.]]] [[1,6],[2],[3],[4],[5]]=>[5,4,3,2,1,6]=>[2,3,4,5,1,6]=>[[.,.],[.,[.,[.,[.,.]]]]] [[1,5],[2],[3],[4],[6]]=>[6,4,3,2,1,5]=>[2,3,4,6,1,5]=>[[.,.],[.,[.,[[.,.],.]]]] [[1,4],[2],[3],[5],[6]]=>[6,5,3,2,1,4]=>[2,3,5,1,6,4]=>[[.,.],[.,[[.,.],[.,.]]]] [[1,3],[2],[4],[5],[6]]=>[6,5,4,2,1,3]=>[2,4,1,5,6,3]=>[[.,.],[[.,.],[.,[.,.]]]] [[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[3,1,4,5,6,2]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]=>[[.,.],[.,[.,[.,[.,.]]]]] [[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]] [[1,3,4,5,6,7],[2]]=>[2,1,3,4,5,6,7]=>[2,1,3,4,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,2,4,5,6,7],[3]]=>[3,1,2,4,5,6,7]=>[3,1,2,4,5,6,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,3,5,6,7],[4]]=>[4,1,2,3,5,6,7]=>[4,1,2,3,5,6,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,3,4,6,7],[5]]=>[5,1,2,3,4,6,7]=>[5,1,2,3,4,6,7]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]] [[1,2,3,4,5,7],[6]]=>[6,1,2,3,4,5,7]=>[6,1,2,3,4,5,7]=>[[.,[.,[.,[.,[.,.]]]]],[.,.]] [[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[7,1,2,3,4,5,6]=>[[.,[.,[.,[.,[.,[.,.]]]]]],.] [[1,3,5,6,7],[2,4]]=>[2,4,1,3,5,6,7]=>[4,2,1,3,5,6,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,5,6,7],[3,4]]=>[3,4,1,2,5,6,7]=>[4,1,3,2,5,6,7]=>[[.,[[.,.],.]],[.,[.,[.,.]]]] [[1,3,4,6,7],[2,5]]=>[2,5,1,3,4,6,7]=>[5,2,1,3,4,6,7]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,4,6,7],[3,5]]=>[3,5,1,2,4,6,7]=>[5,1,3,2,4,6,7]=>[[.,[[.,.],[.,.]]],[.,[.,.]]] [[1,2,3,6,7],[4,5]]=>[4,5,1,2,3,6,7]=>[5,1,2,4,3,6,7]=>[[.,[.,[[.,.],.]]],[.,[.,.]]] [[1,3,4,5,7],[2,6]]=>[2,6,1,3,4,5,7]=>[6,2,1,3,4,5,7]=>[[[.,.],[.,[.,[.,.]]]],[.,.]] [[1,2,4,5,7],[3,6]]=>[3,6,1,2,4,5,7]=>[6,1,3,2,4,5,7]=>[[.,[[.,.],[.,[.,.]]]],[.,.]] [[1,2,3,5,7],[4,6]]=>[4,6,1,2,3,5,7]=>[6,1,2,4,3,5,7]=>[[.,[.,[[.,.],[.,.]]]],[.,.]] [[1,2,3,4,7],[5,6]]=>[5,6,1,2,3,4,7]=>[6,1,2,3,5,4,7]=>[[.,[.,[.,[[.,.],.]]]],[.,.]] [[1,3,4,5,6],[2,7]]=>[2,7,1,3,4,5,6]=>[7,2,1,3,4,5,6]=>[[[.,.],[.,[.,[.,[.,.]]]]],.] [[1,2,4,5,6],[3,7]]=>[3,7,1,2,4,5,6]=>[7,1,3,2,4,5,6]=>[[.,[[.,.],[.,[.,[.,.]]]]],.] [[1,2,3,5,6],[4,7]]=>[4,7,1,2,3,5,6]=>[7,1,2,4,3,5,6]=>[[.,[.,[[.,.],[.,[.,.]]]]],.] [[1,2,3,4,6],[5,7]]=>[5,7,1,2,3,4,6]=>[7,1,2,3,5,4,6]=>[[.,[.,[.,[[.,.],[.,.]]]]],.] [[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[7,1,2,3,4,6,5]=>[[.,[.,[.,[.,[[.,.],.]]]]],.] [[1,4,5,6,7],[2],[3]]=>[3,2,1,4,5,6,7]=>[2,3,1,4,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,3,5,6,7],[2],[4]]=>[4,2,1,3,5,6,7]=>[2,4,1,3,5,6,7]=>[[.,.],[[.,.],[.,[.,[.,.]]]]] [[1,2,5,6,7],[3],[4]]=>[4,3,1,2,5,6,7]=>[3,1,4,2,5,6,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,3,4,6,7],[2],[5]]=>[5,2,1,3,4,6,7]=>[2,5,1,3,4,6,7]=>[[.,.],[[.,[.,.]],[.,[.,.]]]] [[1,2,4,6,7],[3],[5]]=>[5,3,1,2,4,6,7]=>[3,1,5,2,4,6,7]=>[[.,[.,.]],[[.,.],[.,[.,.]]]] [[1,2,3,6,7],[4],[5]]=>[5,4,1,2,3,6,7]=>[4,1,2,5,3,6,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,3,4,5,7],[2],[6]]=>[6,2,1,3,4,5,7]=>[2,6,1,3,4,5,7]=>[[.,.],[[.,[.,[.,.]]],[.,.]]] [[1,2,4,5,7],[3],[6]]=>[6,3,1,2,4,5,7]=>[3,1,6,2,4,5,7]=>[[.,[.,.]],[[.,[.,.]],[.,.]]] [[1,2,3,5,7],[4],[6]]=>[6,4,1,2,3,5,7]=>[4,1,2,6,3,5,7]=>[[.,[.,[.,.]]],[[.,.],[.,.]]] [[1,2,3,4,7],[5],[6]]=>[6,5,1,2,3,4,7]=>[5,1,2,3,6,4,7]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]] [[1,3,4,5,6],[2],[7]]=>[7,2,1,3,4,5,6]=>[2,7,1,3,4,5,6]=>[[.,.],[[.,[.,[.,[.,.]]]],.]] [[1,2,4,5,6],[3],[7]]=>[7,3,1,2,4,5,6]=>[3,1,7,2,4,5,6]=>[[.,[.,.]],[[.,[.,[.,.]]],.]] [[1,2,3,5,6],[4],[7]]=>[7,4,1,2,3,5,6]=>[4,1,2,7,3,5,6]=>[[.,[.,[.,.]]],[[.,[.,.]],.]] [[1,2,3,4,6],[5],[7]]=>[7,5,1,2,3,4,6]=>[5,1,2,3,7,4,6]=>[[.,[.,[.,[.,.]]]],[[.,.],.]] [[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[6,1,2,3,4,7,5]=>[[.,[.,[.,[.,[.,.]]]]],[.,.]] [[1,3,5,7],[2,4,6]]=>[2,4,6,1,3,5,7]=>[6,2,1,4,3,5,7]=>[[[.,.],[[.,.],[.,.]]],[.,.]] [[1,2,5,7],[3,4,6]]=>[3,4,6,1,2,5,7]=>[6,1,3,4,2,5,7]=>[[.,[[.,.],[.,[.,.]]]],[.,.]] [[1,3,4,7],[2,5,6]]=>[2,5,6,1,3,4,7]=>[6,2,1,3,5,4,7]=>[[[.,.],[.,[[.,.],.]]],[.,.]] [[1,2,4,7],[3,5,6]]=>[3,5,6,1,2,4,7]=>[6,1,3,2,5,4,7]=>[[.,[[.,.],[[.,.],.]]],[.,.]] [[1,2,3,7],[4,5,6]]=>[4,5,6,1,2,3,7]=>[6,1,2,4,5,3,7]=>[[.,[.,[[.,.],[.,.]]]],[.,.]] [[1,3,5,6],[2,4,7]]=>[2,4,7,1,3,5,6]=>[7,2,1,4,3,5,6]=>[[[.,.],[[.,.],[.,[.,.]]]],.] [[1,2,5,6],[3,4,7]]=>[3,4,7,1,2,5,6]=>[7,1,3,4,2,5,6]=>[[.,[[.,.],[.,[.,[.,.]]]]],.] [[1,3,4,6],[2,5,7]]=>[2,5,7,1,3,4,6]=>[7,2,1,3,5,4,6]=>[[[.,.],[.,[[.,.],[.,.]]]],.] [[1,2,4,6],[3,5,7]]=>[3,5,7,1,2,4,6]=>[7,1,3,2,5,4,6]=>[[.,[[.,.],[[.,.],[.,.]]]],.] [[1,2,3,6],[4,5,7]]=>[4,5,7,1,2,3,6]=>[7,1,2,4,5,3,6]=>[[.,[.,[[.,.],[.,[.,.]]]]],.] [[1,3,4,5],[2,6,7]]=>[2,6,7,1,3,4,5]=>[7,2,1,3,4,6,5]=>[[[.,.],[.,[.,[[.,.],.]]]],.] [[1,2,4,5],[3,6,7]]=>[3,6,7,1,2,4,5]=>[7,1,3,2,4,6,5]=>[[.,[[.,.],[.,[[.,.],.]]]],.] [[1,2,3,5],[4,6,7]]=>[4,6,7,1,2,3,5]=>[7,1,2,4,3,6,5]=>[[.,[.,[[.,.],[[.,.],.]]]],.] [[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[7,1,2,3,5,6,4]=>[[.,[.,[.,[[.,.],[.,.]]]]],.] [[1,4,6,7],[2,5],[3]]=>[3,2,5,1,4,6,7]=>[5,3,2,1,4,6,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,6,7],[2,5],[4]]=>[4,2,5,1,3,6,7]=>[5,4,1,2,3,6,7]=>[[[.,[.,[.,.]]],.],[.,[.,.]]] [[1,2,6,7],[3,5],[4]]=>[4,3,5,1,2,6,7]=>[5,1,4,3,2,6,7]=>[[.,[[[.,.],.],.]],[.,[.,.]]] [[1,3,6,7],[2,4],[5]]=>[5,2,4,1,3,6,7]=>[4,5,1,2,3,6,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,6,7],[3,4],[5]]=>[5,3,4,1,2,6,7]=>[4,1,5,3,2,6,7]=>[[.,[[.,.],.]],[.,[.,[.,.]]]] [[1,4,5,7],[2,6],[3]]=>[3,2,6,1,4,5,7]=>[6,3,2,1,4,5,7]=>[[[[.,.],.],[.,[.,.]]],[.,.]] [[1,3,5,7],[2,6],[4]]=>[4,2,6,1,3,5,7]=>[6,4,1,2,3,5,7]=>[[[.,[.,[.,.]]],[.,.]],[.,.]] [[1,2,5,7],[3,6],[4]]=>[4,3,6,1,2,5,7]=>[6,1,4,3,2,5,7]=>[[.,[[[.,.],.],[.,.]]],[.,.]] [[1,3,4,7],[2,6],[5]]=>[5,2,6,1,3,4,7]=>[6,5,1,3,2,4,7]=>[[[.,[[.,.],[.,.]]],.],[.,.]] [[1,2,4,7],[3,6],[5]]=>[5,3,6,1,2,4,7]=>[6,1,5,2,3,4,7]=>[[.,[[.,[.,[.,.]]],.]],[.,.]] [[1,2,3,7],[4,6],[5]]=>[5,4,6,1,2,3,7]=>[6,1,2,5,4,3,7]=>[[.,[.,[[[.,.],.],.]]],[.,.]] [[1,3,5,7],[2,4],[6]]=>[6,2,4,1,3,5,7]=>[4,6,1,2,3,5,7]=>[[.,[.,[.,.]]],[[.,.],[.,.]]] [[1,2,5,7],[3,4],[6]]=>[6,3,4,1,2,5,7]=>[4,1,6,3,2,5,7]=>[[.,[[.,.],.]],[[.,.],[.,.]]] [[1,3,4,7],[2,5],[6]]=>[6,2,5,1,3,4,7]=>[5,6,1,3,2,4,7]=>[[.,[[.,.],[.,.]]],[.,[.,.]]] [[1,2,4,7],[3,5],[6]]=>[6,3,5,1,2,4,7]=>[5,1,6,2,3,4,7]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]] [[1,2,3,7],[4,5],[6]]=>[6,4,5,1,2,3,7]=>[5,1,2,6,4,3,7]=>[[.,[.,[[.,.],.]]],[.,[.,.]]] [[1,4,5,6],[2,7],[3]]=>[3,2,7,1,4,5,6]=>[7,3,2,1,4,5,6]=>[[[[.,.],.],[.,[.,[.,.]]]],.] [[1,3,5,6],[2,7],[4]]=>[4,2,7,1,3,5,6]=>[7,4,1,2,3,5,6]=>[[[.,[.,[.,.]]],[.,[.,.]]],.] [[1,2,5,6],[3,7],[4]]=>[4,3,7,1,2,5,6]=>[7,1,4,3,2,5,6]=>[[.,[[[.,.],.],[.,[.,.]]]],.] [[1,3,4,6],[2,7],[5]]=>[5,2,7,1,3,4,6]=>[7,5,1,3,2,4,6]=>[[[.,[[.,.],[.,.]]],[.,.]],.] [[1,2,4,6],[3,7],[5]]=>[5,3,7,1,2,4,6]=>[7,1,5,2,3,4,6]=>[[.,[[.,[.,[.,.]]],[.,.]]],.] [[1,2,3,6],[4,7],[5]]=>[5,4,7,1,2,3,6]=>[7,1,2,5,4,3,6]=>[[.,[.,[[[.,.],.],[.,.]]]],.] [[1,3,4,5],[2,7],[6]]=>[6,2,7,1,3,4,5]=>[7,6,1,3,4,2,5]=>[[[.,[[.,.],[.,[.,.]]]],.],.] [[1,2,4,5],[3,7],[6]]=>[6,3,7,1,2,4,5]=>[7,1,6,2,4,3,5]=>[[.,[[.,[[.,.],[.,.]]],.]],.] [[1,2,3,5],[4,7],[6]]=>[6,4,7,1,2,3,5]=>[7,1,2,6,3,4,5]=>[[.,[.,[[.,[.,[.,.]]],.]]],.] [[1,2,3,4],[5,7],[6]]=>[6,5,7,1,2,3,4]=>[7,1,2,3,6,5,4]=>[[.,[.,[.,[[[.,.],.],.]]]],.] [[1,3,5,6],[2,4],[7]]=>[7,2,4,1,3,5,6]=>[4,7,1,2,3,5,6]=>[[.,[.,[.,.]]],[[.,[.,.]],.]] [[1,2,5,6],[3,4],[7]]=>[7,3,4,1,2,5,6]=>[4,1,7,3,2,5,6]=>[[.,[[.,.],.]],[[.,[.,.]],.]] [[1,3,4,6],[2,5],[7]]=>[7,2,5,1,3,4,6]=>[5,7,1,3,2,4,6]=>[[.,[[.,.],[.,.]]],[[.,.],.]] [[1,2,4,6],[3,5],[7]]=>[7,3,5,1,2,4,6]=>[5,1,7,2,3,4,6]=>[[.,[.,[.,[.,.]]]],[[.,.],.]] [[1,2,3,6],[4,5],[7]]=>[7,4,5,1,2,3,6]=>[5,1,2,7,4,3,6]=>[[.,[.,[[.,.],.]]],[[.,.],.]] [[1,3,4,5],[2,6],[7]]=>[7,2,6,1,3,4,5]=>[6,7,1,3,4,2,5]=>[[.,[[.,.],[.,[.,.]]]],[.,.]] [[1,2,4,5],[3,6],[7]]=>[7,3,6,1,2,4,5]=>[6,1,7,2,4,3,5]=>[[.,[.,[[.,.],[.,.]]]],[.,.]] [[1,2,3,5],[4,6],[7]]=>[7,4,6,1,2,3,5]=>[6,1,2,7,3,4,5]=>[[.,[.,[.,[.,[.,.]]]]],[.,.]] [[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[6,1,2,3,7,5,4]=>[[.,[.,[.,[[.,.],.]]]],[.,.]] [[1,5,6,7],[2],[3],[4]]=>[4,3,2,1,5,6,7]=>[2,3,4,1,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,4,6,7],[2],[3],[5]]=>[5,3,2,1,4,6,7]=>[2,3,5,1,4,6,7]=>[[.,.],[.,[[.,.],[.,[.,.]]]]] [[1,3,6,7],[2],[4],[5]]=>[5,4,2,1,3,6,7]=>[2,4,1,5,3,6,7]=>[[.,.],[[.,.],[.,[.,[.,.]]]]] [[1,2,6,7],[3],[4],[5]]=>[5,4,3,1,2,6,7]=>[3,1,4,5,2,6,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,4,5,7],[2],[3],[6]]=>[6,3,2,1,4,5,7]=>[2,3,6,1,4,5,7]=>[[.,.],[.,[[.,[.,.]],[.,.]]]] [[1,3,5,7],[2],[4],[6]]=>[6,4,2,1,3,5,7]=>[2,4,1,6,3,5,7]=>[[.,.],[[.,.],[[.,.],[.,.]]]] [[1,2,5,7],[3],[4],[6]]=>[6,4,3,1,2,5,7]=>[3,1,4,6,2,5,7]=>[[.,[.,.]],[.,[[.,.],[.,.]]]] [[1,3,4,7],[2],[5],[6]]=>[6,5,2,1,3,4,7]=>[2,5,1,3,6,4,7]=>[[.,.],[[.,[.,.]],[.,[.,.]]]] [[1,2,4,7],[3],[5],[6]]=>[6,5,3,1,2,4,7]=>[3,1,5,2,6,4,7]=>[[.,[.,.]],[[.,.],[.,[.,.]]]] [[1,2,3,7],[4],[5],[6]]=>[6,5,4,1,2,3,7]=>[4,1,2,5,6,3,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,4,5,6],[2],[3],[7]]=>[7,3,2,1,4,5,6]=>[2,3,7,1,4,5,6]=>[[.,.],[.,[[.,[.,[.,.]]],.]]] [[1,3,5,6],[2],[4],[7]]=>[7,4,2,1,3,5,6]=>[2,4,1,7,3,5,6]=>[[.,.],[[.,.],[[.,[.,.]],.]]] [[1,2,5,6],[3],[4],[7]]=>[7,4,3,1,2,5,6]=>[3,1,4,7,2,5,6]=>[[.,[.,.]],[.,[[.,[.,.]],.]]] [[1,3,4,6],[2],[5],[7]]=>[7,5,2,1,3,4,6]=>[2,5,1,3,7,4,6]=>[[.,.],[[.,[.,.]],[[.,.],.]]] [[1,2,4,6],[3],[5],[7]]=>[7,5,3,1,2,4,6]=>[3,1,5,2,7,4,6]=>[[.,[.,.]],[[.,.],[[.,.],.]]] [[1,2,3,6],[4],[5],[7]]=>[7,5,4,1,2,3,6]=>[4,1,2,5,7,3,6]=>[[.,[.,[.,.]]],[.,[[.,.],.]]] [[1,3,4,5],[2],[6],[7]]=>[7,6,2,1,3,4,5]=>[2,6,1,3,4,7,5]=>[[.,.],[[.,[.,[.,.]]],[.,.]]] [[1,2,4,5],[3],[6],[7]]=>[7,6,3,1,2,4,5]=>[3,1,6,2,4,7,5]=>[[.,[.,.]],[[.,[.,.]],[.,.]]] [[1,2,3,5],[4],[6],[7]]=>[7,6,4,1,2,3,5]=>[4,1,2,6,3,7,5]=>[[.,[.,[.,.]]],[[.,.],[.,.]]] [[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[5,1,2,3,6,7,4]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]] [[1,4,6],[2,5,7],[3]]=>[3,2,5,7,1,4,6]=>[7,3,2,1,5,4,6]=>[[[[.,.],.],[[.,.],[.,.]]],.] [[1,3,6],[2,5,7],[4]]=>[4,2,5,7,1,3,6]=>[7,4,1,2,5,3,6]=>[[[.,[.,[.,.]]],[.,[.,.]]],.] [[1,2,6],[3,5,7],[4]]=>[4,3,5,7,1,2,6]=>[7,1,4,3,5,2,6]=>[[.,[[[.,.],.],[.,[.,.]]]],.] [[1,3,6],[2,4,7],[5]]=>[5,2,4,7,1,3,6]=>[7,5,1,2,4,3,6]=>[[[.,[.,[[.,.],.]]],[.,.]],.] [[1,2,6],[3,4,7],[5]]=>[5,3,4,7,1,2,6]=>[7,1,5,3,4,2,6]=>[[.,[[[.,.],[.,.]],[.,.]]],.] [[1,4,5],[2,6,7],[3]]=>[3,2,6,7,1,4,5]=>[7,3,2,1,4,6,5]=>[[[[.,.],.],[.,[[.,.],.]]],.] [[1,3,5],[2,6,7],[4]]=>[4,2,6,7,1,3,5]=>[7,4,1,2,3,6,5]=>[[[.,[.,[.,.]]],[[.,.],.]],.] [[1,2,5],[3,6,7],[4]]=>[4,3,6,7,1,2,5]=>[7,1,4,3,2,6,5]=>[[.,[[[.,.],.],[[.,.],.]]],.] [[1,3,4],[2,6,7],[5]]=>[5,2,6,7,1,3,4]=>[7,5,1,3,2,6,4]=>[[[.,[[.,.],[.,.]]],[.,.]],.] [[1,2,4],[3,6,7],[5]]=>[5,3,6,7,1,2,4]=>[7,1,5,2,3,6,4]=>[[.,[[.,[.,[.,.]]],[.,.]]],.] [[1,2,3],[4,6,7],[5]]=>[5,4,6,7,1,2,3]=>[7,1,2,5,4,6,3]=>[[.,[.,[[[.,.],.],[.,.]]]],.] [[1,3,5],[2,4,7],[6]]=>[6,2,4,7,1,3,5]=>[7,6,1,2,3,4,5]=>[[[.,[.,[.,[.,[.,.]]]]],.],.] [[1,2,5],[3,4,7],[6]]=>[6,3,4,7,1,2,5]=>[7,1,6,3,2,4,5]=>[[.,[[[.,.],[.,[.,.]]],.]],.] [[1,3,4],[2,5,7],[6]]=>[6,2,5,7,1,3,4]=>[7,6,1,3,2,5,4]=>[[[.,[[.,.],[[.,.],.]]],.],.] [[1,2,4],[3,5,7],[6]]=>[6,3,5,7,1,2,4]=>[7,1,6,2,3,5,4]=>[[.,[[.,[.,[[.,.],.]]],.]],.] [[1,2,3],[4,5,7],[6]]=>[6,4,5,7,1,2,3]=>[7,1,2,6,4,5,3]=>[[.,[.,[[[.,.],[.,.]],.]]],.] [[1,3,5],[2,4,6],[7]]=>[7,2,4,6,1,3,5]=>[6,7,1,2,3,4,5]=>[[.,[.,[.,[.,[.,.]]]]],[.,.]] [[1,2,5],[3,4,6],[7]]=>[7,3,4,6,1,2,5]=>[6,1,7,3,2,4,5]=>[[.,[[.,.],[.,[.,.]]]],[.,.]] [[1,3,4],[2,5,6],[7]]=>[7,2,5,6,1,3,4]=>[6,7,1,3,2,5,4]=>[[.,[[.,.],[[.,.],.]]],[.,.]] [[1,2,4],[3,5,6],[7]]=>[7,3,5,6,1,2,4]=>[6,1,7,2,3,5,4]=>[[.,[.,[.,[[.,.],.]]]],[.,.]] [[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[6,1,2,7,4,5,3]=>[[.,[.,[[.,.],[.,.]]]],[.,.]] [[1,4,7],[2,5],[3,6]]=>[3,6,2,5,1,4,7]=>[5,6,3,1,2,4,7]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3,7],[2,5],[4,6]]=>[4,6,2,5,1,3,7]=>[5,6,1,4,2,3,7]=>[[.,[[.,[.,.]],.]],[.,[.,.]]] [[1,2,7],[3,5],[4,6]]=>[4,6,3,5,1,2,7]=>[5,1,6,4,3,2,7]=>[[.,[[[.,.],.],.]],[.,[.,.]]] [[1,3,7],[2,4],[5,6]]=>[5,6,2,4,1,3,7]=>[4,6,1,2,5,3,7]=>[[.,[.,[.,.]]],[[.,.],[.,.]]] [[1,2,7],[3,4],[5,6]]=>[5,6,3,4,1,2,7]=>[4,1,6,3,5,2,7]=>[[.,[[.,.],.]],[[.,.],[.,.]]] [[1,4,6],[2,5],[3,7]]=>[3,7,2,5,1,4,6]=>[5,7,3,1,2,4,6]=>[[[.,[.,.]],[.,.]],[[.,.],.]] [[1,3,6],[2,5],[4,7]]=>[4,7,2,5,1,3,6]=>[5,7,1,4,2,3,6]=>[[.,[[.,[.,.]],.]],[[.,.],.]] [[1,2,6],[3,5],[4,7]]=>[4,7,3,5,1,2,6]=>[5,1,7,4,3,2,6]=>[[.,[[[.,.],.],.]],[[.,.],.]] [[1,3,6],[2,4],[5,7]]=>[5,7,2,4,1,3,6]=>[4,7,1,2,5,3,6]=>[[.,[.,[.,.]]],[[.,[.,.]],.]] [[1,2,6],[3,4],[5,7]]=>[5,7,3,4,1,2,6]=>[4,1,7,3,5,2,6]=>[[.,[[.,.],.]],[[.,[.,.]],.]] [[1,4,5],[2,6],[3,7]]=>[3,7,2,6,1,4,5]=>[6,7,3,1,4,2,5]=>[[[.,[.,.]],[.,[.,.]]],[.,.]] [[1,3,5],[2,6],[4,7]]=>[4,7,2,6,1,3,5]=>[6,7,1,4,3,2,5]=>[[.,[[[.,.],.],[.,.]]],[.,.]] [[1,2,5],[3,6],[4,7]]=>[4,7,3,6,1,2,5]=>[6,1,7,4,2,3,5]=>[[.,[[.,[.,.]],[.,.]]],[.,.]] [[1,3,4],[2,6],[5,7]]=>[5,7,2,6,1,3,4]=>[6,7,1,3,5,2,4]=>[[.,[[.,.],[[.,.],.]]],[.,.]] [[1,2,4],[3,6],[5,7]]=>[5,7,3,6,1,2,4]=>[6,1,7,2,5,3,4]=>[[.,[.,[[.,[.,.]],.]]],[.,.]] [[1,2,3],[4,6],[5,7]]=>[5,7,4,6,1,2,3]=>[6,1,2,7,5,4,3]=>[[.,[.,[[[.,.],.],.]]],[.,.]] [[1,3,5],[2,4],[6,7]]=>[6,7,2,4,1,3,5]=>[4,7,1,2,3,6,5]=>[[.,[.,[.,.]]],[[[.,.],.],.]] [[1,2,5],[3,4],[6,7]]=>[6,7,3,4,1,2,5]=>[4,1,7,3,2,6,5]=>[[.,[[.,.],.]],[[[.,.],.],.]] [[1,3,4],[2,5],[6,7]]=>[6,7,2,5,1,3,4]=>[5,7,1,3,2,6,4]=>[[.,[[.,.],[.,.]]],[[.,.],.]] [[1,2,4],[3,5],[6,7]]=>[6,7,3,5,1,2,4]=>[5,1,7,2,3,6,4]=>[[.,[.,[.,[.,.]]]],[[.,.],.]] [[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[5,1,2,7,4,6,3]=>[[.,[.,[[.,.],.]]],[[.,.],.]] [[1,5,7],[2,6],[3],[4]]=>[4,3,2,6,1,5,7]=>[6,3,4,2,1,5,7]=>[[[[.,.],.],[.,[.,.]]],[.,.]] [[1,4,7],[2,6],[3],[5]]=>[5,3,2,6,1,4,7]=>[6,3,5,1,2,4,7]=>[[[.,[.,.]],[[.,.],.]],[.,.]] [[1,3,7],[2,6],[4],[5]]=>[5,4,2,6,1,3,7]=>[6,4,1,5,2,3,7]=>[[[.,[.,[.,.]]],[.,.]],[.,.]] [[1,2,7],[3,6],[4],[5]]=>[5,4,3,6,1,2,7]=>[6,1,4,5,3,2,7]=>[[.,[[[.,.],.],[.,.]]],[.,.]] [[1,4,7],[2,5],[3],[6]]=>[6,3,2,5,1,4,7]=>[5,3,6,1,2,4,7]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3,7],[2,5],[4],[6]]=>[6,4,2,5,1,3,7]=>[5,4,1,6,2,3,7]=>[[[.,[.,[.,.]]],.],[.,[.,.]]] [[1,2,7],[3,5],[4],[6]]=>[6,4,3,5,1,2,7]=>[5,1,4,6,3,2,7]=>[[.,[[[.,.],.],.]],[.,[.,.]]] [[1,3,7],[2,4],[5],[6]]=>[6,5,2,4,1,3,7]=>[4,5,1,2,6,3,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,7],[3,4],[5],[6]]=>[6,5,3,4,1,2,7]=>[4,1,5,3,6,2,7]=>[[.,[[.,.],.]],[.,[.,[.,.]]]] [[1,5,6],[2,7],[3],[4]]=>[4,3,2,7,1,5,6]=>[7,3,4,2,1,5,6]=>[[[[.,.],.],[.,[.,[.,.]]]],.] [[1,4,6],[2,7],[3],[5]]=>[5,3,2,7,1,4,6]=>[7,3,5,1,2,4,6]=>[[[.,[.,.]],[[.,.],[.,.]]],.] [[1,3,6],[2,7],[4],[5]]=>[5,4,2,7,1,3,6]=>[7,4,1,5,2,3,6]=>[[[.,[.,[.,.]]],[.,[.,.]]],.] [[1,2,6],[3,7],[4],[5]]=>[5,4,3,7,1,2,6]=>[7,1,4,5,3,2,6]=>[[.,[[[.,.],.],[.,[.,.]]]],.] [[1,4,5],[2,7],[3],[6]]=>[6,3,2,7,1,4,5]=>[7,3,6,1,4,2,5]=>[[[.,[.,.]],[[.,[.,.]],.]],.] [[1,3,5],[2,7],[4],[6]]=>[6,4,2,7,1,3,5]=>[7,4,1,6,3,2,5]=>[[[.,[[.,.],.]],[[.,.],.]],.] [[1,2,5],[3,7],[4],[6]]=>[6,4,3,7,1,2,5]=>[7,1,4,6,2,3,5]=>[[.,[[.,[.,.]],[[.,.],.]]],.] [[1,3,4],[2,7],[5],[6]]=>[6,5,2,7,1,3,4]=>[7,5,1,3,6,2,4]=>[[[.,[[.,.],[.,.]]],[.,.]],.] [[1,2,4],[3,7],[5],[6]]=>[6,5,3,7,1,2,4]=>[7,1,5,2,6,3,4]=>[[.,[[.,[.,[.,.]]],[.,.]]],.] [[1,2,3],[4,7],[5],[6]]=>[6,5,4,7,1,2,3]=>[7,1,2,5,6,4,3]=>[[.,[.,[[[.,.],.],[.,.]]]],.] [[1,4,6],[2,5],[3],[7]]=>[7,3,2,5,1,4,6]=>[5,3,7,1,2,4,6]=>[[[.,[.,.]],[.,.]],[[.,.],.]] [[1,3,6],[2,5],[4],[7]]=>[7,4,2,5,1,3,6]=>[5,4,1,7,2,3,6]=>[[[.,[.,[.,.]]],.],[[.,.],.]] [[1,2,6],[3,5],[4],[7]]=>[7,4,3,5,1,2,6]=>[5,1,4,7,3,2,6]=>[[.,[[[.,.],.],.]],[[.,.],.]] [[1,3,6],[2,4],[5],[7]]=>[7,5,2,4,1,3,6]=>[4,5,1,2,7,3,6]=>[[.,[.,[.,.]]],[.,[[.,.],.]]] [[1,2,6],[3,4],[5],[7]]=>[7,5,3,4,1,2,6]=>[4,1,5,3,7,2,6]=>[[.,[[.,.],.]],[.,[[.,.],.]]] [[1,4,5],[2,6],[3],[7]]=>[7,3,2,6,1,4,5]=>[6,3,7,1,4,2,5]=>[[[.,[.,.]],[.,[.,.]]],[.,.]] [[1,3,5],[2,6],[4],[7]]=>[7,4,2,6,1,3,5]=>[6,4,1,7,3,2,5]=>[[[.,[[.,.],.]],[.,.]],[.,.]] [[1,2,5],[3,6],[4],[7]]=>[7,4,3,6,1,2,5]=>[6,1,4,7,2,3,5]=>[[.,[[.,[.,.]],[.,.]]],[.,.]] [[1,3,4],[2,6],[5],[7]]=>[7,5,2,6,1,3,4]=>[6,5,1,3,7,2,4]=>[[[.,[[.,.],[.,.]]],.],[.,.]] [[1,2,4],[3,6],[5],[7]]=>[7,5,3,6,1,2,4]=>[6,1,5,2,7,3,4]=>[[.,[[.,[.,[.,.]]],.]],[.,.]] [[1,2,3],[4,6],[5],[7]]=>[7,5,4,6,1,2,3]=>[6,1,2,5,7,4,3]=>[[.,[.,[[[.,.],.],.]]],[.,.]] [[1,3,5],[2,4],[6],[7]]=>[7,6,2,4,1,3,5]=>[4,6,1,2,3,7,5]=>[[.,[.,[.,.]]],[[.,.],[.,.]]] [[1,2,5],[3,4],[6],[7]]=>[7,6,3,4,1,2,5]=>[4,1,6,3,2,7,5]=>[[.,[[.,.],.]],[[.,.],[.,.]]] [[1,3,4],[2,5],[6],[7]]=>[7,6,2,5,1,3,4]=>[5,6,1,3,2,7,4]=>[[.,[[.,.],[.,.]]],[.,[.,.]]] [[1,2,4],[3,5],[6],[7]]=>[7,6,3,5,1,2,4]=>[5,1,6,2,3,7,4]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]] [[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[5,1,2,6,4,7,3]=>[[.,[.,[[.,.],.]]],[.,[.,.]]] [[1,6,7],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7]=>[2,3,4,5,1,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,5,7],[2],[3],[4],[6]]=>[6,4,3,2,1,5,7]=>[2,3,4,6,1,5,7]=>[[.,.],[.,[.,[[.,.],[.,.]]]]] [[1,4,7],[2],[3],[5],[6]]=>[6,5,3,2,1,4,7]=>[2,3,5,1,6,4,7]=>[[.,.],[.,[[.,.],[.,[.,.]]]]] [[1,3,7],[2],[4],[5],[6]]=>[6,5,4,2,1,3,7]=>[2,4,1,5,6,3,7]=>[[.,.],[[.,.],[.,[.,[.,.]]]]] [[1,2,7],[3],[4],[5],[6]]=>[6,5,4,3,1,2,7]=>[3,1,4,5,6,2,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,5,6],[2],[3],[4],[7]]=>[7,4,3,2,1,5,6]=>[2,3,4,7,1,5,6]=>[[.,.],[.,[.,[[.,[.,.]],.]]]] [[1,4,6],[2],[3],[5],[7]]=>[7,5,3,2,1,4,6]=>[2,3,5,1,7,4,6]=>[[.,.],[.,[[.,.],[[.,.],.]]]] [[1,3,6],[2],[4],[5],[7]]=>[7,5,4,2,1,3,6]=>[2,4,1,5,7,3,6]=>[[.,.],[[.,.],[.,[[.,.],.]]]] [[1,2,6],[3],[4],[5],[7]]=>[7,5,4,3,1,2,6]=>[3,1,4,5,7,2,6]=>[[.,[.,.]],[.,[.,[[.,.],.]]]] [[1,4,5],[2],[3],[6],[7]]=>[7,6,3,2,1,4,5]=>[2,3,6,1,4,7,5]=>[[.,.],[.,[[.,[.,.]],[.,.]]]] [[1,3,5],[2],[4],[6],[7]]=>[7,6,4,2,1,3,5]=>[2,4,1,6,3,7,5]=>[[.,.],[[.,.],[[.,.],[.,.]]]] [[1,2,5],[3],[4],[6],[7]]=>[7,6,4,3,1,2,5]=>[3,1,4,6,2,7,5]=>[[.,[.,.]],[.,[[.,.],[.,.]]]] [[1,3,4],[2],[5],[6],[7]]=>[7,6,5,2,1,3,4]=>[2,5,1,3,6,7,4]=>[[.,.],[[.,[.,.]],[.,[.,.]]]] [[1,2,4],[3],[5],[6],[7]]=>[7,6,5,3,1,2,4]=>[3,1,5,2,6,7,4]=>[[.,[.,.]],[[.,.],[.,[.,.]]]] [[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[4,1,2,5,6,7,3]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,5],[2,6],[3,7],[4]]=>[4,3,7,2,6,1,5]=>[6,7,4,3,1,2,5]=>[[[[.,[.,.]],.],[.,.]],[.,.]] [[1,4],[2,6],[3,7],[5]]=>[5,3,7,2,6,1,4]=>[6,7,5,1,3,2,4]=>[[[.,[[.,.],[.,.]]],.],[.,.]] [[1,3],[2,6],[4,7],[5]]=>[5,4,7,2,6,1,3]=>[6,7,1,5,4,2,3]=>[[.,[[[.,[.,.]],.],.]],[.,.]] [[1,2],[3,6],[4,7],[5]]=>[5,4,7,3,6,1,2]=>[6,1,7,5,4,3,2]=>[[.,[[[[.,.],.],.],.]],[.,.]] [[1,4],[2,5],[3,7],[6]]=>[6,3,7,2,5,1,4]=>[5,7,6,1,2,3,4]=>[[.,[.,[.,[.,.]]]],[[.,.],.]] [[1,3],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3]=>[5,7,1,6,2,4,3]=>[[.,[.,[[.,.],.]]],[[.,.],.]] [[1,2],[3,5],[4,7],[6]]=>[6,4,7,3,5,1,2]=>[5,1,7,6,3,4,2]=>[[.,[[.,.],[.,.]]],[[.,.],.]] [[1,3],[2,4],[5,7],[6]]=>[6,5,7,2,4,1,3]=>[4,7,1,2,6,5,3]=>[[.,[.,[.,.]]],[[[.,.],.],.]] [[1,2],[3,4],[5,7],[6]]=>[6,5,7,3,4,1,2]=>[4,1,7,3,6,5,2]=>[[.,[[.,.],.]],[[[.,.],.],.]] [[1,4],[2,5],[3,6],[7]]=>[7,3,6,2,5,1,4]=>[5,6,7,1,2,3,4]=>[[.,[.,[.,[.,.]]]],[.,[.,.]]] [[1,3],[2,5],[4,6],[7]]=>[7,4,6,2,5,1,3]=>[5,6,1,7,2,4,3]=>[[.,[.,[[.,.],.]]],[.,[.,.]]] [[1,2],[3,5],[4,6],[7]]=>[7,4,6,3,5,1,2]=>[5,1,6,7,3,4,2]=>[[.,[[.,.],[.,.]]],[.,[.,.]]] [[1,3],[2,4],[5,6],[7]]=>[7,5,6,2,4,1,3]=>[4,6,1,2,7,5,3]=>[[.,[.,[.,.]]],[[.,.],[.,.]]] [[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[4,1,6,3,7,5,2]=>[[.,[[.,.],.]],[[.,.],[.,.]]] [[1,6],[2,7],[3],[4],[5]]=>[5,4,3,2,7,1,6]=>[7,3,4,5,2,1,6]=>[[[[.,.],.],[.,[.,[.,.]]]],.] [[1,5],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5]=>[7,3,4,6,1,2,5]=>[[[.,[.,.]],[.,[[.,.],.]]],.] [[1,4],[2,7],[3],[5],[6]]=>[6,5,3,2,7,1,4]=>[7,3,5,1,6,2,4]=>[[[.,[.,.]],[[.,.],[.,.]]],.] [[1,3],[2,7],[4],[5],[6]]=>[6,5,4,2,7,1,3]=>[7,4,1,5,6,2,3]=>[[[.,[.,[.,.]]],[.,[.,.]]],.] [[1,2],[3,7],[4],[5],[6]]=>[6,5,4,3,7,1,2]=>[7,1,4,5,6,3,2]=>[[.,[[[.,.],.],[.,[.,.]]]],.] [[1,5],[2,6],[3],[4],[7]]=>[7,4,3,2,6,1,5]=>[6,3,4,7,1,2,5]=>[[[.,[.,.]],[.,[.,.]]],[.,.]] [[1,4],[2,6],[3],[5],[7]]=>[7,5,3,2,6,1,4]=>[6,3,5,1,7,2,4]=>[[[.,[.,.]],[[.,.],.]],[.,.]] [[1,3],[2,6],[4],[5],[7]]=>[7,5,4,2,6,1,3]=>[6,4,1,5,7,2,3]=>[[[.,[.,[.,.]]],[.,.]],[.,.]] [[1,2],[3,6],[4],[5],[7]]=>[7,5,4,3,6,1,2]=>[6,1,4,5,7,3,2]=>[[.,[[[.,.],.],[.,.]]],[.,.]] [[1,4],[2,5],[3],[6],[7]]=>[7,6,3,2,5,1,4]=>[5,3,6,1,2,7,4]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3],[2,5],[4],[6],[7]]=>[7,6,4,2,5,1,3]=>[5,4,1,6,2,7,3]=>[[[.,[.,[.,.]]],.],[.,[.,.]]] [[1,2],[3,5],[4],[6],[7]]=>[7,6,4,3,5,1,2]=>[5,1,4,6,3,7,2]=>[[.,[[[.,.],.],.]],[.,[.,.]]] [[1,3],[2,4],[5],[6],[7]]=>[7,6,5,2,4,1,3]=>[4,5,1,2,6,7,3]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[4,1,5,3,6,7,2]=>[[.,[[.,.],.]],[.,[.,[.,.]]]] [[1,7],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7]=>[2,3,4,5,6,1,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,6],[2],[3],[4],[5],[7]]=>[7,5,4,3,2,1,6]=>[2,3,4,5,7,1,6]=>[[.,.],[.,[.,[.,[[.,.],.]]]]] [[1,5],[2],[3],[4],[6],[7]]=>[7,6,4,3,2,1,5]=>[2,3,4,6,1,7,5]=>[[.,.],[.,[.,[[.,.],[.,.]]]]] [[1,4],[2],[3],[5],[6],[7]]=>[7,6,5,3,2,1,4]=>[2,3,5,1,6,7,4]=>[[.,.],[.,[[.,.],[.,[.,.]]]]] [[1,3],[2],[4],[5],[6],[7]]=>[7,6,5,4,2,1,3]=>[2,4,1,5,6,7,3]=>[[.,.],[[.,.],[.,[.,[.,.]]]]] [[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[3,1,4,5,6,7,2]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[2,3,4,5,6,7,1]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8]=>[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
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
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
  • the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
  • the set of left-to-right maximima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
  • the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
  • the set of maximal elements in the decreasing runs of $\pi$ is the set of deficiency positions of $\chi(\pi)$, and
  • the set of minimal elements in the decreasing runs of $\pi$ is the set of deficiency values of $\chi(\pi)$.
Map
binary search tree: left to right
Description
Return the shape of the binary search tree of the permutation as a non labelled binary tree.