Identifier
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Images
=>
Cc0007;cc-rep-0Cc0007;cc-rep-1Cc0010;cc-rep-3
[[1]]=>[[1]]=>[1]=>[.,.] [[1,2]]=>[[1,2]]=>[1,2]=>[.,[.,.]] [[1],[2]]=>[[1],[2]]=>[2,1]=>[[.,.],.] [[1,2,3]]=>[[1,2,3]]=>[1,2,3]=>[.,[.,[.,.]]] [[1,3],[2]]=>[[1,2],[3]]=>[3,1,2]=>[[.,.],[.,.]] [[1,2],[3]]=>[[1,3],[2]]=>[2,1,3]=>[[.,.],[.,.]] [[1],[2],[3]]=>[[1],[2],[3]]=>[3,2,1]=>[[[.,.],.],.] [[1,2,3,4]]=>[[1,2,3,4]]=>[1,2,3,4]=>[.,[.,[.,[.,.]]]] [[1,3,4],[2]]=>[[1,2,3],[4]]=>[4,1,2,3]=>[[.,.],[.,[.,.]]] [[1,2,4],[3]]=>[[1,2,4],[3]]=>[3,1,2,4]=>[[.,.],[.,[.,.]]] [[1,2,3],[4]]=>[[1,3,4],[2]]=>[2,1,3,4]=>[[.,.],[.,[.,.]]] [[1,3],[2,4]]=>[[1,3],[2,4]]=>[2,4,1,3]=>[[.,[.,.]],[.,.]] [[1,2],[3,4]]=>[[1,2],[3,4]]=>[3,4,1,2]=>[[.,[.,.]],[.,.]] [[1,4],[2],[3]]=>[[1,2],[3],[4]]=>[4,3,1,2]=>[[[.,.],.],[.,.]] [[1,3],[2],[4]]=>[[1,3],[2],[4]]=>[4,2,1,3]=>[[[.,.],.],[.,.]] [[1,2],[3],[4]]=>[[1,4],[2],[3]]=>[3,2,1,4]=>[[[.,.],.],[.,.]] [[1],[2],[3],[4]]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[[[[.,.],.],.],.] [[1,2,3,4,5]]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>[.,[.,[.,[.,[.,.]]]]] [[1,3,4,5],[2]]=>[[1,2,3,4],[5]]=>[5,1,2,3,4]=>[[.,.],[.,[.,[.,.]]]] [[1,2,4,5],[3]]=>[[1,2,3,5],[4]]=>[4,1,2,3,5]=>[[.,.],[.,[.,[.,.]]]] [[1,2,3,5],[4]]=>[[1,2,4,5],[3]]=>[3,1,2,4,5]=>[[.,.],[.,[.,[.,.]]]] [[1,2,3,4],[5]]=>[[1,3,4,5],[2]]=>[2,1,3,4,5]=>[[.,.],[.,[.,[.,.]]]] [[1,3,5],[2,4]]=>[[1,2,4],[3,5]]=>[3,5,1,2,4]=>[[.,[.,.]],[.,[.,.]]] [[1,2,5],[3,4]]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[[.,[.,.]],[.,[.,.]]] [[1,3,4],[2,5]]=>[[1,3,4],[2,5]]=>[2,5,1,3,4]=>[[.,[.,.]],[.,[.,.]]] [[1,2,4],[3,5]]=>[[1,3,5],[2,4]]=>[2,4,1,3,5]=>[[.,[.,.]],[.,[.,.]]] [[1,2,3],[4,5]]=>[[1,2,5],[3,4]]=>[3,4,1,2,5]=>[[.,[.,.]],[.,[.,.]]] [[1,4,5],[2],[3]]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[[[.,.],.],[.,[.,.]]] [[1,3,5],[2],[4]]=>[[1,2,4],[3],[5]]=>[5,3,1,2,4]=>[[[.,.],.],[.,[.,.]]] [[1,2,5],[3],[4]]=>[[1,2,5],[3],[4]]=>[4,3,1,2,5]=>[[[.,.],.],[.,[.,.]]] [[1,3,4],[2],[5]]=>[[1,3,4],[2],[5]]=>[5,2,1,3,4]=>[[[.,.],.],[.,[.,.]]] [[1,2,4],[3],[5]]=>[[1,3,5],[2],[4]]=>[4,2,1,3,5]=>[[[.,.],.],[.,[.,.]]] [[1,2,3],[4],[5]]=>[[1,4,5],[2],[3]]=>[3,2,1,4,5]=>[[[.,.],.],[.,[.,.]]] [[1,4],[2,5],[3]]=>[[1,3],[2,4],[5]]=>[5,2,4,1,3]=>[[[.,.],[.,.]],[.,.]] [[1,3],[2,5],[4]]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[[[.,.],[.,.]],[.,.]] [[1,2],[3,5],[4]]=>[[1,2],[3,5],[4]]=>[4,3,5,1,2]=>[[[.,.],[.,.]],[.,.]] [[1,3],[2,4],[5]]=>[[1,4],[2,5],[3]]=>[3,2,5,1,4]=>[[[.,.],[.,.]],[.,.]] [[1,2],[3,4],[5]]=>[[1,3],[2,5],[4]]=>[4,2,5,1,3]=>[[[.,.],[.,.]],[.,.]] [[1,5],[2],[3],[4]]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[[[[.,.],.],.],[.,.]] [[1,4],[2],[3],[5]]=>[[1,3],[2],[4],[5]]=>[5,4,2,1,3]=>[[[[.,.],.],.],[.,.]] [[1,3],[2],[4],[5]]=>[[1,4],[2],[3],[5]]=>[5,3,2,1,4]=>[[[[.,.],.],.],[.,.]] [[1,2],[3],[4],[5]]=>[[1,5],[2],[3],[4]]=>[4,3,2,1,5]=>[[[[.,.],.],.],[.,.]] [[1],[2],[3],[4],[5]]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[[[[[.,.],.],.],.],.] [[1,2,3,4,5,6]]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[.,[.,[.,[.,[.,[.,.]]]]]] [[1,3,4,5,6],[2]]=>[[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[[.,.],[.,[.,[.,[.,.]]]]] [[1,2,4,5,6],[3]]=>[[1,2,3,4,6],[5]]=>[5,1,2,3,4,6]=>[[.,.],[.,[.,[.,[.,.]]]]] [[1,2,3,5,6],[4]]=>[[1,2,3,5,6],[4]]=>[4,1,2,3,5,6]=>[[.,.],[.,[.,[.,[.,.]]]]] [[1,2,3,4,6],[5]]=>[[1,2,4,5,6],[3]]=>[3,1,2,4,5,6]=>[[.,.],[.,[.,[.,[.,.]]]]] [[1,2,3,4,5],[6]]=>[[1,3,4,5,6],[2]]=>[2,1,3,4,5,6]=>[[.,.],[.,[.,[.,[.,.]]]]] [[1,3,5,6],[2,4]]=>[[1,2,3,5],[4,6]]=>[4,6,1,2,3,5]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,2,5,6],[3,4]]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,3,4,6],[2,5]]=>[[1,2,4,5],[3,6]]=>[3,6,1,2,4,5]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,2,4,6],[3,5]]=>[[1,2,4,6],[3,5]]=>[3,5,1,2,4,6]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,2,3,6],[4,5]]=>[[1,2,3,6],[4,5]]=>[4,5,1,2,3,6]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,3,4,5],[2,6]]=>[[1,3,4,5],[2,6]]=>[2,6,1,3,4,5]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,2,4,5],[3,6]]=>[[1,3,4,6],[2,5]]=>[2,5,1,3,4,6]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,2,3,5],[4,6]]=>[[1,3,5,6],[2,4]]=>[2,4,1,3,5,6]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,2,3,4],[5,6]]=>[[1,2,5,6],[3,4]]=>[3,4,1,2,5,6]=>[[.,[.,.]],[.,[.,[.,.]]]] [[1,4,5,6],[2],[3]]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[[[.,.],.],[.,[.,[.,.]]]] [[1,3,5,6],[2],[4]]=>[[1,2,3,5],[4],[6]]=>[6,4,1,2,3,5]=>[[[.,.],.],[.,[.,[.,.]]]] [[1,2,5,6],[3],[4]]=>[[1,2,3,6],[4],[5]]=>[5,4,1,2,3,6]=>[[[.,.],.],[.,[.,[.,.]]]] [[1,3,4,6],[2],[5]]=>[[1,2,4,5],[3],[6]]=>[6,3,1,2,4,5]=>[[[.,.],.],[.,[.,[.,.]]]] [[1,2,4,6],[3],[5]]=>[[1,2,4,6],[3],[5]]=>[5,3,1,2,4,6]=>[[[.,.],.],[.,[.,[.,.]]]] [[1,2,3,6],[4],[5]]=>[[1,2,5,6],[3],[4]]=>[4,3,1,2,5,6]=>[[[.,.],.],[.,[.,[.,.]]]] [[1,3,4,5],[2],[6]]=>[[1,3,4,5],[2],[6]]=>[6,2,1,3,4,5]=>[[[.,.],.],[.,[.,[.,.]]]] [[1,2,4,5],[3],[6]]=>[[1,3,4,6],[2],[5]]=>[5,2,1,3,4,6]=>[[[.,.],.],[.,[.,[.,.]]]] [[1,2,3,5],[4],[6]]=>[[1,3,5,6],[2],[4]]=>[4,2,1,3,5,6]=>[[[.,.],.],[.,[.,[.,.]]]] [[1,2,3,4],[5],[6]]=>[[1,4,5,6],[2],[3]]=>[3,2,1,4,5,6]=>[[[.,.],.],[.,[.,[.,.]]]] [[1,3,5],[2,4,6]]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[[.,[.,[.,.]]],[.,[.,.]]] [[1,2,5],[3,4,6]]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[[.,[.,[.,.]]],[.,[.,.]]] [[1,3,4],[2,5,6]]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[[.,[.,[.,.]]],[.,[.,.]]] [[1,2,4],[3,5,6]]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[[.,[.,[.,.]]],[.,[.,.]]] [[1,2,3],[4,5,6]]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[[.,[.,[.,.]]],[.,[.,.]]] [[1,4,6],[2,5],[3]]=>[[1,2,4],[3,5],[6]]=>[6,3,5,1,2,4]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,3,6],[2,5],[4]]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,2,6],[3,5],[4]]=>[[1,2,3],[4,6],[5]]=>[5,4,6,1,2,3]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,3,6],[2,4],[5]]=>[[1,2,5],[3,6],[4]]=>[4,3,6,1,2,5]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,2,6],[3,4],[5]]=>[[1,2,4],[3,6],[5]]=>[5,3,6,1,2,4]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,4,5],[2,6],[3]]=>[[1,3,4],[2,5],[6]]=>[6,2,5,1,3,4]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,3,5],[2,6],[4]]=>[[1,3,5],[2,4],[6]]=>[6,2,4,1,3,5]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,2,5],[3,6],[4]]=>[[1,3,6],[2,4],[5]]=>[5,2,4,1,3,6]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,3,4],[2,6],[5]]=>[[1,2,5],[3,4],[6]]=>[6,3,4,1,2,5]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,2,4],[3,6],[5]]=>[[1,2,6],[3,4],[5]]=>[5,3,4,1,2,6]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,2,3],[4,6],[5]]=>[[1,2,6],[3,5],[4]]=>[4,3,5,1,2,6]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,3,5],[2,4],[6]]=>[[1,3,5],[2,6],[4]]=>[4,2,6,1,3,5]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,2,5],[3,4],[6]]=>[[1,3,4],[2,6],[5]]=>[5,2,6,1,3,4]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,3,4],[2,5],[6]]=>[[1,4,5],[2,6],[3]]=>[3,2,6,1,4,5]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,2,4],[3,5],[6]]=>[[1,4,6],[2,5],[3]]=>[3,2,5,1,4,6]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,2,3],[4,5],[6]]=>[[1,3,6],[2,5],[4]]=>[4,2,5,1,3,6]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,5,6],[2],[3],[4]]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[[[[.,.],.],.],[.,[.,.]]] [[1,4,6],[2],[3],[5]]=>[[1,2,4],[3],[5],[6]]=>[6,5,3,1,2,4]=>[[[[.,.],.],.],[.,[.,.]]] [[1,3,6],[2],[4],[5]]=>[[1,2,5],[3],[4],[6]]=>[6,4,3,1,2,5]=>[[[[.,.],.],.],[.,[.,.]]] [[1,2,6],[3],[4],[5]]=>[[1,2,6],[3],[4],[5]]=>[5,4,3,1,2,6]=>[[[[.,.],.],.],[.,[.,.]]] [[1,4,5],[2],[3],[6]]=>[[1,3,4],[2],[5],[6]]=>[6,5,2,1,3,4]=>[[[[.,.],.],.],[.,[.,.]]] [[1,3,5],[2],[4],[6]]=>[[1,3,5],[2],[4],[6]]=>[6,4,2,1,3,5]=>[[[[.,.],.],.],[.,[.,.]]] [[1,2,5],[3],[4],[6]]=>[[1,3,6],[2],[4],[5]]=>[5,4,2,1,3,6]=>[[[[.,.],.],.],[.,[.,.]]] [[1,3,4],[2],[5],[6]]=>[[1,4,5],[2],[3],[6]]=>[6,3,2,1,4,5]=>[[[[.,.],.],.],[.,[.,.]]] [[1,2,4],[3],[5],[6]]=>[[1,4,6],[2],[3],[5]]=>[5,3,2,1,4,6]=>[[[[.,.],.],.],[.,[.,.]]] [[1,2,3],[4],[5],[6]]=>[[1,5,6],[2],[3],[4]]=>[4,3,2,1,5,6]=>[[[[.,.],.],.],[.,[.,.]]] [[1,4],[2,5],[3,6]]=>[[1,4],[2,5],[3,6]]=>[3,6,2,5,1,4]=>[[[.,[.,.]],[.,.]],[.,.]] [[1,3],[2,5],[4,6]]=>[[1,3],[2,5],[4,6]]=>[4,6,2,5,1,3]=>[[[.,[.,.]],[.,.]],[.,.]] [[1,2],[3,5],[4,6]]=>[[1,3],[2,4],[5,6]]=>[5,6,2,4,1,3]=>[[[.,[.,.]],[.,.]],[.,.]] [[1,3],[2,4],[5,6]]=>[[1,2],[3,5],[4,6]]=>[4,6,3,5,1,2]=>[[[.,[.,.]],[.,.]],[.,.]] [[1,2],[3,4],[5,6]]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[[[.,[.,.]],[.,.]],[.,.]] [[1,5],[2,6],[3],[4]]=>[[1,3],[2,4],[5],[6]]=>[6,5,2,4,1,3]=>[[[[.,.],.],[.,.]],[.,.]] [[1,4],[2,6],[3],[5]]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[[[[.,.],.],[.,.]],[.,.]] [[1,3],[2,6],[4],[5]]=>[[1,2],[3,5],[4],[6]]=>[6,4,3,5,1,2]=>[[[[.,.],.],[.,.]],[.,.]] [[1,2],[3,6],[4],[5]]=>[[1,2],[3,6],[4],[5]]=>[5,4,3,6,1,2]=>[[[[.,.],.],[.,.]],[.,.]] [[1,4],[2,5],[3],[6]]=>[[1,4],[2,5],[3],[6]]=>[6,3,2,5,1,4]=>[[[[.,.],.],[.,.]],[.,.]] [[1,3],[2,5],[4],[6]]=>[[1,3],[2,5],[4],[6]]=>[6,4,2,5,1,3]=>[[[[.,.],.],[.,.]],[.,.]] [[1,2],[3,5],[4],[6]]=>[[1,3],[2,6],[4],[5]]=>[5,4,2,6,1,3]=>[[[[.,.],.],[.,.]],[.,.]] [[1,3],[2,4],[5],[6]]=>[[1,5],[2,6],[3],[4]]=>[4,3,2,6,1,5]=>[[[[.,.],.],[.,.]],[.,.]] [[1,2],[3,4],[5],[6]]=>[[1,4],[2,6],[3],[5]]=>[5,3,2,6,1,4]=>[[[[.,.],.],[.,.]],[.,.]] [[1,6],[2],[3],[4],[5]]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[[[[[.,.],.],.],.],[.,.]] [[1,5],[2],[3],[4],[6]]=>[[1,3],[2],[4],[5],[6]]=>[6,5,4,2,1,3]=>[[[[[.,.],.],.],.],[.,.]] [[1,4],[2],[3],[5],[6]]=>[[1,4],[2],[3],[5],[6]]=>[6,5,3,2,1,4]=>[[[[[.,.],.],.],.],[.,.]] [[1,3],[2],[4],[5],[6]]=>[[1,5],[2],[3],[4],[6]]=>[6,4,3,2,1,5]=>[[[[[.,.],.],.],.],[.,.]] [[1,2],[3],[4],[5],[6]]=>[[1,6],[2],[3],[4],[5]]=>[5,4,3,2,1,6]=>[[[[[.,.],.],.],.],[.,.]] [[1],[2],[3],[4],[5],[6]]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[[[[[[.,.],.],.],.],.],.] [[1,2,3,4,5,6,7]]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[.,[.,[.,[.,[.,[.,[.,.]]]]]]] [[1,3,4,5,6,7],[2]]=>[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,2,4,5,6,7],[3]]=>[[1,2,3,4,5,7],[6]]=>[6,1,2,3,4,5,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,2,3,5,6,7],[4]]=>[[1,2,3,4,6,7],[5]]=>[5,1,2,3,4,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,2,3,4,6,7],[5]]=>[[1,2,3,5,6,7],[4]]=>[4,1,2,3,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,2,3,4,5,7],[6]]=>[[1,2,4,5,6,7],[3]]=>[3,1,2,4,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,2,3,4,5,6],[7]]=>[[1,3,4,5,6,7],[2]]=>[2,1,3,4,5,6,7]=>[[.,.],[.,[.,[.,[.,[.,.]]]]]] [[1,3,5,6,7],[2,4]]=>[[1,2,3,4,6],[5,7]]=>[5,7,1,2,3,4,6]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,5,6,7],[3,4]]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,3,4,6,7],[2,5]]=>[[1,2,3,5,6],[4,7]]=>[4,7,1,2,3,5,6]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,4,6,7],[3,5]]=>[[1,2,3,5,7],[4,6]]=>[4,6,1,2,3,5,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,3,6,7],[4,5]]=>[[1,2,3,4,7],[5,6]]=>[5,6,1,2,3,4,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,3,4,5,7],[2,6]]=>[[1,2,4,5,6],[3,7]]=>[3,7,1,2,4,5,6]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,4,5,7],[3,6]]=>[[1,2,4,5,7],[3,6]]=>[3,6,1,2,4,5,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,3,5,7],[4,6]]=>[[1,2,4,6,7],[3,5]]=>[3,5,1,2,4,6,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,3,4,7],[5,6]]=>[[1,2,3,6,7],[4,5]]=>[4,5,1,2,3,6,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,3,4,5,6],[2,7]]=>[[1,3,4,5,6],[2,7]]=>[2,7,1,3,4,5,6]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,4,5,6],[3,7]]=>[[1,3,4,5,7],[2,6]]=>[2,6,1,3,4,5,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,3,5,6],[4,7]]=>[[1,3,4,6,7],[2,5]]=>[2,5,1,3,4,6,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,3,4,6],[5,7]]=>[[1,3,5,6,7],[2,4]]=>[2,4,1,3,5,6,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,2,3,4,5],[6,7]]=>[[1,2,5,6,7],[3,4]]=>[3,4,1,2,5,6,7]=>[[.,[.,.]],[.,[.,[.,[.,.]]]]] [[1,4,5,6,7],[2],[3]]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,3,5,6,7],[2],[4]]=>[[1,2,3,4,6],[5],[7]]=>[7,5,1,2,3,4,6]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,2,5,6,7],[3],[4]]=>[[1,2,3,4,7],[5],[6]]=>[6,5,1,2,3,4,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,3,4,6,7],[2],[5]]=>[[1,2,3,5,6],[4],[7]]=>[7,4,1,2,3,5,6]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,2,4,6,7],[3],[5]]=>[[1,2,3,5,7],[4],[6]]=>[6,4,1,2,3,5,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,2,3,6,7],[4],[5]]=>[[1,2,3,6,7],[4],[5]]=>[5,4,1,2,3,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,3,4,5,7],[2],[6]]=>[[1,2,4,5,6],[3],[7]]=>[7,3,1,2,4,5,6]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,2,4,5,7],[3],[6]]=>[[1,2,4,5,7],[3],[6]]=>[6,3,1,2,4,5,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,2,3,5,7],[4],[6]]=>[[1,2,4,6,7],[3],[5]]=>[5,3,1,2,4,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,2,3,4,7],[5],[6]]=>[[1,2,5,6,7],[3],[4]]=>[4,3,1,2,5,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,3,4,5,6],[2],[7]]=>[[1,3,4,5,6],[2],[7]]=>[7,2,1,3,4,5,6]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,2,4,5,6],[3],[7]]=>[[1,3,4,5,7],[2],[6]]=>[6,2,1,3,4,5,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,2,3,5,6],[4],[7]]=>[[1,3,4,6,7],[2],[5]]=>[5,2,1,3,4,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,2,3,4,6],[5],[7]]=>[[1,3,5,6,7],[2],[4]]=>[4,2,1,3,5,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,2,3,4,5],[6],[7]]=>[[1,4,5,6,7],[2],[3]]=>[3,2,1,4,5,6,7]=>[[[.,.],.],[.,[.,[.,[.,.]]]]] [[1,3,5,7],[2,4,6]]=>[[1,2,4,6],[3,5,7]]=>[3,5,7,1,2,4,6]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,5,7],[3,4,6]]=>[[1,2,4,5],[3,6,7]]=>[3,6,7,1,2,4,5]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,3,4,7],[2,5,6]]=>[[1,2,3,6],[4,5,7]]=>[4,5,7,1,2,3,6]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,4,7],[3,5,6]]=>[[1,2,3,5],[4,6,7]]=>[4,6,7,1,2,3,5]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,3,7],[4,5,6]]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,3,5,6],[2,4,7]]=>[[1,3,4,6],[2,5,7]]=>[2,5,7,1,3,4,6]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,5,6],[3,4,7]]=>[[1,3,4,5],[2,6,7]]=>[2,6,7,1,3,4,5]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,3,4,6],[2,5,7]]=>[[1,3,5,6],[2,4,7]]=>[2,4,7,1,3,5,6]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,4,6],[3,5,7]]=>[[1,3,5,7],[2,4,6]]=>[2,4,6,1,3,5,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,3,6],[4,5,7]]=>[[1,3,4,7],[2,5,6]]=>[2,5,6,1,3,4,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,3,4,5],[2,6,7]]=>[[1,2,5,6],[3,4,7]]=>[3,4,7,1,2,5,6]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,4,5],[3,6,7]]=>[[1,2,5,7],[3,4,6]]=>[3,4,6,1,2,5,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,3,5],[4,6,7]]=>[[1,2,4,7],[3,5,6]]=>[3,5,6,1,2,4,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,2,3,4],[5,6,7]]=>[[1,2,3,7],[4,5,6]]=>[4,5,6,1,2,3,7]=>[[.,[.,[.,.]]],[.,[.,[.,.]]]] [[1,4,6,7],[2,5],[3]]=>[[1,2,3,5],[4,6],[7]]=>[7,4,6,1,2,3,5]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,6,7],[2,5],[4]]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,6,7],[3,5],[4]]=>[[1,2,3,4],[5,7],[6]]=>[6,5,7,1,2,3,4]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,6,7],[2,4],[5]]=>[[1,2,3,6],[4,7],[5]]=>[5,4,7,1,2,3,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,6,7],[3,4],[5]]=>[[1,2,3,5],[4,7],[6]]=>[6,4,7,1,2,3,5]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,4,5,7],[2,6],[3]]=>[[1,2,4,5],[3,6],[7]]=>[7,3,6,1,2,4,5]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,5,7],[2,6],[4]]=>[[1,2,4,6],[3,5],[7]]=>[7,3,5,1,2,4,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,5,7],[3,6],[4]]=>[[1,2,4,7],[3,5],[6]]=>[6,3,5,1,2,4,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,4,7],[2,6],[5]]=>[[1,2,3,6],[4,5],[7]]=>[7,4,5,1,2,3,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,4,7],[3,6],[5]]=>[[1,2,3,7],[4,5],[6]]=>[6,4,5,1,2,3,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,3,7],[4,6],[5]]=>[[1,2,3,7],[4,6],[5]]=>[5,4,6,1,2,3,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,5,7],[2,4],[6]]=>[[1,2,4,6],[3,7],[5]]=>[5,3,7,1,2,4,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,5,7],[3,4],[6]]=>[[1,2,4,5],[3,7],[6]]=>[6,3,7,1,2,4,5]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,4,7],[2,5],[6]]=>[[1,2,5,6],[3,7],[4]]=>[4,3,7,1,2,5,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,4,7],[3,5],[6]]=>[[1,2,5,7],[3,6],[4]]=>[4,3,6,1,2,5,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,3,7],[4,5],[6]]=>[[1,2,4,7],[3,6],[5]]=>[5,3,6,1,2,4,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,4,5,6],[2,7],[3]]=>[[1,3,4,5],[2,6],[7]]=>[7,2,6,1,3,4,5]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,5,6],[2,7],[4]]=>[[1,3,4,6],[2,5],[7]]=>[7,2,5,1,3,4,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,5,6],[3,7],[4]]=>[[1,3,4,7],[2,5],[6]]=>[6,2,5,1,3,4,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,4,6],[2,7],[5]]=>[[1,3,5,6],[2,4],[7]]=>[7,2,4,1,3,5,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,4,6],[3,7],[5]]=>[[1,3,5,7],[2,4],[6]]=>[6,2,4,1,3,5,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,3,6],[4,7],[5]]=>[[1,3,6,7],[2,4],[5]]=>[5,2,4,1,3,6,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,4,5],[2,7],[6]]=>[[1,2,5,6],[3,4],[7]]=>[7,3,4,1,2,5,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,4,5],[3,7],[6]]=>[[1,2,5,7],[3,4],[6]]=>[6,3,4,1,2,5,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,3,5],[4,7],[6]]=>[[1,2,6,7],[3,4],[5]]=>[5,3,4,1,2,6,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,3,4],[5,7],[6]]=>[[1,2,6,7],[3,5],[4]]=>[4,3,5,1,2,6,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,5,6],[2,4],[7]]=>[[1,3,4,6],[2,7],[5]]=>[5,2,7,1,3,4,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,5,6],[3,4],[7]]=>[[1,3,4,5],[2,7],[6]]=>[6,2,7,1,3,4,5]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,4,6],[2,5],[7]]=>[[1,3,5,6],[2,7],[4]]=>[4,2,7,1,3,5,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,4,6],[3,5],[7]]=>[[1,3,5,7],[2,6],[4]]=>[4,2,6,1,3,5,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,3,6],[4,5],[7]]=>[[1,3,4,7],[2,6],[5]]=>[5,2,6,1,3,4,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,3,4,5],[2,6],[7]]=>[[1,4,5,6],[2,7],[3]]=>[3,2,7,1,4,5,6]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,4,5],[3,6],[7]]=>[[1,4,5,7],[2,6],[3]]=>[3,2,6,1,4,5,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,3,5],[4,6],[7]]=>[[1,4,6,7],[2,5],[3]]=>[3,2,5,1,4,6,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,2,3,4],[5,6],[7]]=>[[1,3,6,7],[2,5],[4]]=>[4,2,5,1,3,6,7]=>[[[.,.],[.,.]],[.,[.,[.,.]]]] [[1,5,6,7],[2],[3],[4]]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,4,6,7],[2],[3],[5]]=>[[1,2,3,5],[4],[6],[7]]=>[7,6,4,1,2,3,5]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,3,6,7],[2],[4],[5]]=>[[1,2,3,6],[4],[5],[7]]=>[7,5,4,1,2,3,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,2,6,7],[3],[4],[5]]=>[[1,2,3,7],[4],[5],[6]]=>[6,5,4,1,2,3,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,4,5,7],[2],[3],[6]]=>[[1,2,4,5],[3],[6],[7]]=>[7,6,3,1,2,4,5]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,3,5,7],[2],[4],[6]]=>[[1,2,4,6],[3],[5],[7]]=>[7,5,3,1,2,4,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,2,5,7],[3],[4],[6]]=>[[1,2,4,7],[3],[5],[6]]=>[6,5,3,1,2,4,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,3,4,7],[2],[5],[6]]=>[[1,2,5,6],[3],[4],[7]]=>[7,4,3,1,2,5,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,2,4,7],[3],[5],[6]]=>[[1,2,5,7],[3],[4],[6]]=>[6,4,3,1,2,5,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,2,3,7],[4],[5],[6]]=>[[1,2,6,7],[3],[4],[5]]=>[5,4,3,1,2,6,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,4,5,6],[2],[3],[7]]=>[[1,3,4,5],[2],[6],[7]]=>[7,6,2,1,3,4,5]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,3,5,6],[2],[4],[7]]=>[[1,3,4,6],[2],[5],[7]]=>[7,5,2,1,3,4,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,2,5,6],[3],[4],[7]]=>[[1,3,4,7],[2],[5],[6]]=>[6,5,2,1,3,4,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,3,4,6],[2],[5],[7]]=>[[1,3,5,6],[2],[4],[7]]=>[7,4,2,1,3,5,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,2,4,6],[3],[5],[7]]=>[[1,3,5,7],[2],[4],[6]]=>[6,4,2,1,3,5,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,2,3,6],[4],[5],[7]]=>[[1,3,6,7],[2],[4],[5]]=>[5,4,2,1,3,6,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,3,4,5],[2],[6],[7]]=>[[1,4,5,6],[2],[3],[7]]=>[7,3,2,1,4,5,6]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,2,4,5],[3],[6],[7]]=>[[1,4,5,7],[2],[3],[6]]=>[6,3,2,1,4,5,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,2,3,5],[4],[6],[7]]=>[[1,4,6,7],[2],[3],[5]]=>[5,3,2,1,4,6,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,2,3,4],[5],[6],[7]]=>[[1,5,6,7],[2],[3],[4]]=>[4,3,2,1,5,6,7]=>[[[[.,.],.],.],[.,[.,[.,.]]]] [[1,4,6],[2,5,7],[3]]=>[[1,3,5],[2,4,6],[7]]=>[7,2,4,6,1,3,5]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,3,6],[2,5,7],[4]]=>[[1,3,4],[2,5,6],[7]]=>[7,2,5,6,1,3,4]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,6],[3,5,7],[4]]=>[[1,3,4],[2,5,7],[6]]=>[6,2,5,7,1,3,4]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,3,6],[2,4,7],[5]]=>[[1,3,6],[2,4,7],[5]]=>[5,2,4,7,1,3,6]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,6],[3,4,7],[5]]=>[[1,3,5],[2,4,7],[6]]=>[6,2,4,7,1,3,5]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,4,5],[2,6,7],[3]]=>[[1,2,5],[3,4,6],[7]]=>[7,3,4,6,1,2,5]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,3,5],[2,6,7],[4]]=>[[1,2,4],[3,5,6],[7]]=>[7,3,5,6,1,2,4]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,5],[3,6,7],[4]]=>[[1,2,4],[3,5,7],[6]]=>[6,3,5,7,1,2,4]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,3,4],[2,6,7],[5]]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,4],[3,6,7],[5]]=>[[1,2,3],[4,5,7],[6]]=>[6,4,5,7,1,2,3]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,3],[4,6,7],[5]]=>[[1,2,3],[4,6,7],[5]]=>[5,4,6,7,1,2,3]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,3,5],[2,4,7],[6]]=>[[1,2,6],[3,4,7],[5]]=>[5,3,4,7,1,2,6]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,5],[3,4,7],[6]]=>[[1,2,5],[3,4,7],[6]]=>[6,3,4,7,1,2,5]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,3,4],[2,5,7],[6]]=>[[1,2,6],[3,5,7],[4]]=>[4,3,5,7,1,2,6]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,4],[3,5,7],[6]]=>[[1,2,5],[3,6,7],[4]]=>[4,3,6,7,1,2,5]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,3],[4,5,7],[6]]=>[[1,2,4],[3,6,7],[5]]=>[5,3,6,7,1,2,4]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,3,5],[2,4,6],[7]]=>[[1,4,6],[2,5,7],[3]]=>[3,2,5,7,1,4,6]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,5],[3,4,6],[7]]=>[[1,4,5],[2,6,7],[3]]=>[3,2,6,7,1,4,5]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,3,4],[2,5,6],[7]]=>[[1,3,6],[2,5,7],[4]]=>[4,2,5,7,1,3,6]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,4],[3,5,6],[7]]=>[[1,3,5],[2,6,7],[4]]=>[4,2,6,7,1,3,5]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,2,3],[4,5,6],[7]]=>[[1,3,4],[2,6,7],[5]]=>[5,2,6,7,1,3,4]=>[[[.,.],[.,[.,.]]],[.,[.,.]]] [[1,4,7],[2,5],[3,6]]=>[[1,2,5],[3,6],[4,7]]=>[4,7,3,6,1,2,5]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3,7],[2,5],[4,6]]=>[[1,2,4],[3,6],[5,7]]=>[5,7,3,6,1,2,4]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,2,7],[3,5],[4,6]]=>[[1,2,4],[3,5],[6,7]]=>[6,7,3,5,1,2,4]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3,7],[2,4],[5,6]]=>[[1,2,3],[4,6],[5,7]]=>[5,7,4,6,1,2,3]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,2,7],[3,4],[5,6]]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,4,6],[2,5],[3,7]]=>[[1,3,5],[2,6],[4,7]]=>[4,7,2,6,1,3,5]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3,6],[2,5],[4,7]]=>[[1,3,4],[2,6],[5,7]]=>[5,7,2,6,1,3,4]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,2,6],[3,5],[4,7]]=>[[1,3,4],[2,5],[6,7]]=>[6,7,2,5,1,3,4]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3,6],[2,4],[5,7]]=>[[1,3,6],[2,4],[5,7]]=>[5,7,2,4,1,3,6]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,2,6],[3,4],[5,7]]=>[[1,3,5],[2,4],[6,7]]=>[6,7,2,4,1,3,5]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,4,5],[2,6],[3,7]]=>[[1,4,5],[2,6],[3,7]]=>[3,7,2,6,1,4,5]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3,5],[2,6],[4,7]]=>[[1,4,6],[2,5],[3,7]]=>[3,7,2,5,1,4,6]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,2,5],[3,6],[4,7]]=>[[1,4,7],[2,5],[3,6]]=>[3,6,2,5,1,4,7]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3,4],[2,6],[5,7]]=>[[1,3,6],[2,5],[4,7]]=>[4,7,2,5,1,3,6]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,2,4],[3,6],[5,7]]=>[[1,3,7],[2,5],[4,6]]=>[4,6,2,5,1,3,7]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,2,3],[4,6],[5,7]]=>[[1,3,7],[2,4],[5,6]]=>[5,6,2,4,1,3,7]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3,5],[2,4],[6,7]]=>[[1,2,6],[3,4],[5,7]]=>[5,7,3,4,1,2,6]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,2,5],[3,4],[6,7]]=>[[1,2,5],[3,4],[6,7]]=>[6,7,3,4,1,2,5]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,3,4],[2,5],[6,7]]=>[[1,2,6],[3,5],[4,7]]=>[4,7,3,5,1,2,6]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,2,4],[3,5],[6,7]]=>[[1,2,7],[3,5],[4,6]]=>[4,6,3,5,1,2,7]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,2,3],[4,5],[6,7]]=>[[1,2,7],[3,4],[5,6]]=>[5,6,3,4,1,2,7]=>[[[.,[.,.]],[.,.]],[.,[.,.]]] [[1,5,7],[2,6],[3],[4]]=>[[1,2,4],[3,5],[6],[7]]=>[7,6,3,5,1,2,4]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,4,7],[2,6],[3],[5]]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,7],[2,6],[4],[5]]=>[[1,2,3],[4,6],[5],[7]]=>[7,5,4,6,1,2,3]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,7],[3,6],[4],[5]]=>[[1,2,3],[4,7],[5],[6]]=>[6,5,4,7,1,2,3]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,4,7],[2,5],[3],[6]]=>[[1,2,5],[3,6],[4],[7]]=>[7,4,3,6,1,2,5]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,7],[2,5],[4],[6]]=>[[1,2,4],[3,6],[5],[7]]=>[7,5,3,6,1,2,4]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,7],[3,5],[4],[6]]=>[[1,2,4],[3,7],[5],[6]]=>[6,5,3,7,1,2,4]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,7],[2,4],[5],[6]]=>[[1,2,6],[3,7],[4],[5]]=>[5,4,3,7,1,2,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,7],[3,4],[5],[6]]=>[[1,2,5],[3,7],[4],[6]]=>[6,4,3,7,1,2,5]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,5,6],[2,7],[3],[4]]=>[[1,3,4],[2,5],[6],[7]]=>[7,6,2,5,1,3,4]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,4,6],[2,7],[3],[5]]=>[[1,3,5],[2,4],[6],[7]]=>[7,6,2,4,1,3,5]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,6],[2,7],[4],[5]]=>[[1,3,6],[2,4],[5],[7]]=>[7,5,2,4,1,3,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,6],[3,7],[4],[5]]=>[[1,3,7],[2,4],[5],[6]]=>[6,5,2,4,1,3,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,4,5],[2,7],[3],[6]]=>[[1,2,5],[3,4],[6],[7]]=>[7,6,3,4,1,2,5]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,5],[2,7],[4],[6]]=>[[1,2,6],[3,4],[5],[7]]=>[7,5,3,4,1,2,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,5],[3,7],[4],[6]]=>[[1,2,7],[3,4],[5],[6]]=>[6,5,3,4,1,2,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,4],[2,7],[5],[6]]=>[[1,2,6],[3,5],[4],[7]]=>[7,4,3,5,1,2,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,4],[3,7],[5],[6]]=>[[1,2,7],[3,5],[4],[6]]=>[6,4,3,5,1,2,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,3],[4,7],[5],[6]]=>[[1,2,7],[3,6],[4],[5]]=>[5,4,3,6,1,2,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,4,6],[2,5],[3],[7]]=>[[1,3,5],[2,6],[4],[7]]=>[7,4,2,6,1,3,5]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,6],[2,5],[4],[7]]=>[[1,3,4],[2,6],[5],[7]]=>[7,5,2,6,1,3,4]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,6],[3,5],[4],[7]]=>[[1,3,4],[2,7],[5],[6]]=>[6,5,2,7,1,3,4]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,6],[2,4],[5],[7]]=>[[1,3,6],[2,7],[4],[5]]=>[5,4,2,7,1,3,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,6],[3,4],[5],[7]]=>[[1,3,5],[2,7],[4],[6]]=>[6,4,2,7,1,3,5]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,4,5],[2,6],[3],[7]]=>[[1,4,5],[2,6],[3],[7]]=>[7,3,2,6,1,4,5]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,5],[2,6],[4],[7]]=>[[1,4,6],[2,5],[3],[7]]=>[7,3,2,5,1,4,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,5],[3,6],[4],[7]]=>[[1,4,7],[2,5],[3],[6]]=>[6,3,2,5,1,4,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,4],[2,6],[5],[7]]=>[[1,3,6],[2,5],[4],[7]]=>[7,4,2,5,1,3,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,4],[3,6],[5],[7]]=>[[1,3,7],[2,5],[4],[6]]=>[6,4,2,5,1,3,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,3],[4,6],[5],[7]]=>[[1,3,7],[2,6],[4],[5]]=>[5,4,2,6,1,3,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,5],[2,4],[6],[7]]=>[[1,4,6],[2,7],[3],[5]]=>[5,3,2,7,1,4,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,5],[3,4],[6],[7]]=>[[1,4,5],[2,7],[3],[6]]=>[6,3,2,7,1,4,5]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,3,4],[2,5],[6],[7]]=>[[1,5,6],[2,7],[3],[4]]=>[4,3,2,7,1,5,6]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,4],[3,5],[6],[7]]=>[[1,5,7],[2,6],[3],[4]]=>[4,3,2,6,1,5,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,2,3],[4,5],[6],[7]]=>[[1,4,7],[2,6],[3],[5]]=>[5,3,2,6,1,4,7]=>[[[[.,.],.],[.,.]],[.,[.,.]]] [[1,6,7],[2],[3],[4],[5]]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,5,7],[2],[3],[4],[6]]=>[[1,2,4],[3],[5],[6],[7]]=>[7,6,5,3,1,2,4]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,4,7],[2],[3],[5],[6]]=>[[1,2,5],[3],[4],[6],[7]]=>[7,6,4,3,1,2,5]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,3,7],[2],[4],[5],[6]]=>[[1,2,6],[3],[4],[5],[7]]=>[7,5,4,3,1,2,6]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,2,7],[3],[4],[5],[6]]=>[[1,2,7],[3],[4],[5],[6]]=>[6,5,4,3,1,2,7]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,5,6],[2],[3],[4],[7]]=>[[1,3,4],[2],[5],[6],[7]]=>[7,6,5,2,1,3,4]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,4,6],[2],[3],[5],[7]]=>[[1,3,5],[2],[4],[6],[7]]=>[7,6,4,2,1,3,5]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,3,6],[2],[4],[5],[7]]=>[[1,3,6],[2],[4],[5],[7]]=>[7,5,4,2,1,3,6]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,2,6],[3],[4],[5],[7]]=>[[1,3,7],[2],[4],[5],[6]]=>[6,5,4,2,1,3,7]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,4,5],[2],[3],[6],[7]]=>[[1,4,5],[2],[3],[6],[7]]=>[7,6,3,2,1,4,5]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,3,5],[2],[4],[6],[7]]=>[[1,4,6],[2],[3],[5],[7]]=>[7,5,3,2,1,4,6]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,2,5],[3],[4],[6],[7]]=>[[1,4,7],[2],[3],[5],[6]]=>[6,5,3,2,1,4,7]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,3,4],[2],[5],[6],[7]]=>[[1,5,6],[2],[3],[4],[7]]=>[7,4,3,2,1,5,6]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,2,4],[3],[5],[6],[7]]=>[[1,5,7],[2],[3],[4],[6]]=>[6,4,3,2,1,5,7]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,2,3],[4],[5],[6],[7]]=>[[1,6,7],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7]=>[[[[[.,.],.],.],.],[.,[.,.]]] [[1,5],[2,6],[3,7],[4]]=>[[1,4],[2,5],[3,6],[7]]=>[7,3,6,2,5,1,4]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,4],[2,6],[3,7],[5]]=>[[1,3],[2,5],[4,6],[7]]=>[7,4,6,2,5,1,3]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,3],[2,6],[4,7],[5]]=>[[1,3],[2,4],[5,6],[7]]=>[7,5,6,2,4,1,3]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,2],[3,6],[4,7],[5]]=>[[1,3],[2,4],[5,7],[6]]=>[6,5,7,2,4,1,3]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,4],[2,5],[3,7],[6]]=>[[1,2],[3,5],[4,6],[7]]=>[7,4,6,3,5,1,2]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,3],[2,5],[4,7],[6]]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,2],[3,5],[4,7],[6]]=>[[1,2],[3,4],[5,7],[6]]=>[6,5,7,3,4,1,2]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,3],[2,4],[5,7],[6]]=>[[1,2],[3,6],[4,7],[5]]=>[5,4,7,3,6,1,2]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,2],[3,4],[5,7],[6]]=>[[1,2],[3,5],[4,7],[6]]=>[6,4,7,3,5,1,2]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,4],[2,5],[3,6],[7]]=>[[1,5],[2,6],[3,7],[4]]=>[4,3,7,2,6,1,5]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,3],[2,5],[4,6],[7]]=>[[1,4],[2,6],[3,7],[5]]=>[5,3,7,2,6,1,4]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,2],[3,5],[4,6],[7]]=>[[1,4],[2,5],[3,7],[6]]=>[6,3,7,2,5,1,4]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,3],[2,4],[5,6],[7]]=>[[1,3],[2,6],[4,7],[5]]=>[5,4,7,2,6,1,3]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,2],[3,4],[5,6],[7]]=>[[1,3],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3]=>[[[[.,.],[.,.]],[.,.]],[.,.]] [[1,6],[2,7],[3],[4],[5]]=>[[1,3],[2,4],[5],[6],[7]]=>[7,6,5,2,4,1,3]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,5],[2,7],[3],[4],[6]]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,4],[2,7],[3],[5],[6]]=>[[1,2],[3,5],[4],[6],[7]]=>[7,6,4,3,5,1,2]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,3],[2,7],[4],[5],[6]]=>[[1,2],[3,6],[4],[5],[7]]=>[7,5,4,3,6,1,2]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,2],[3,7],[4],[5],[6]]=>[[1,2],[3,7],[4],[5],[6]]=>[6,5,4,3,7,1,2]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,5],[2,6],[3],[4],[7]]=>[[1,4],[2,5],[3],[6],[7]]=>[7,6,3,2,5,1,4]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,4],[2,6],[3],[5],[7]]=>[[1,3],[2,5],[4],[6],[7]]=>[7,6,4,2,5,1,3]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,3],[2,6],[4],[5],[7]]=>[[1,3],[2,6],[4],[5],[7]]=>[7,5,4,2,6,1,3]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,2],[3,6],[4],[5],[7]]=>[[1,3],[2,7],[4],[5],[6]]=>[6,5,4,2,7,1,3]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,4],[2,5],[3],[6],[7]]=>[[1,5],[2,6],[3],[4],[7]]=>[7,4,3,2,6,1,5]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,3],[2,5],[4],[6],[7]]=>[[1,4],[2,6],[3],[5],[7]]=>[7,5,3,2,6,1,4]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,2],[3,5],[4],[6],[7]]=>[[1,4],[2,7],[3],[5],[6]]=>[6,5,3,2,7,1,4]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,3],[2,4],[5],[6],[7]]=>[[1,6],[2,7],[3],[4],[5]]=>[5,4,3,2,7,1,6]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,2],[3,4],[5],[6],[7]]=>[[1,5],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5]=>[[[[[.,.],.],.],[.,.]],[.,.]] [[1,7],[2],[3],[4],[5],[6]]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[[[[[[.,.],.],.],.],.],[.,.]] [[1,6],[2],[3],[4],[5],[7]]=>[[1,3],[2],[4],[5],[6],[7]]=>[7,6,5,4,2,1,3]=>[[[[[[.,.],.],.],.],.],[.,.]] [[1,5],[2],[3],[4],[6],[7]]=>[[1,4],[2],[3],[5],[6],[7]]=>[7,6,5,3,2,1,4]=>[[[[[[.,.],.],.],.],.],[.,.]] [[1,4],[2],[3],[5],[6],[7]]=>[[1,5],[2],[3],[4],[6],[7]]=>[7,6,4,3,2,1,5]=>[[[[[[.,.],.],.],.],.],[.,.]] [[1,3],[2],[4],[5],[6],[7]]=>[[1,6],[2],[3],[4],[5],[7]]=>[7,5,4,3,2,1,6]=>[[[[[[.,.],.],.],.],.],[.,.]] [[1,2],[3],[4],[5],[6],[7]]=>[[1,7],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7]=>[[[[[[.,.],.],.],.],.],[.,.]] [[1],[2],[3],[4],[5],[6],[7]]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[[[[[[[.,.],.],.],.],.],.],.] [[1,2,3,4,5,6,7,8]]=>[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]] [[1,2,5,6,7,8],[3,4]]=>[[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]] [[1,2,3,4],[5,6,7,8]]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]] [[1,2,7,8],[3,4],[5,6]]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]] [[1,2,3],[4,5,6],[7],[8]]=>[[1,4,5],[2,7,8],[3],[6]]=>[6,3,2,7,8,1,4,5]=>[[[[.,.],.],[.,[.,.]]],[.,[.,.]]] [[1,2],[3,4],[5,6],[7,8]]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[[[[.,[.,.]],[.,.]],[.,.]],[.,.]] [[1,2],[3,4],[5],[6],[7],[8]]=>[[1,6],[2,8],[3],[4],[5],[7]]=>[7,5,4,3,2,8,1,6]=>[[[[[[.,.],.],.],.],[.,.]],[.,.]] [[1],[2],[3],[4],[5],[6],[7],[8]]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[[[[[[[[.,.],.],.],.],.],.],.],.]
Map
Schützenberger involution
Description
Sends a standard tableau to the standard tableau obtained via the Schützenberger involution.
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
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.