Identifier
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ 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
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.