Identifier
Mp00033: Dyck paths to two-row standard tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00061: Permutations to increasing tree Binary trees
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0010;cc-rep-3
[1,0]=>[[1],[2]]=>[2,1]=>[[.,.],.] [1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>[[.,[.,.]],[.,.]] [1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>[[.,[.,.]],[.,.]] [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[[.,[.,[.,.]]],[.,[.,.]]] [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[[.,[.,[.,.]]],[.,[.,.]]] [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[[.,[.,[.,.]]],[.,[.,.]]] [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[[.,[.,[.,.]]],[.,[.,.]]] [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[[.,[.,[.,.]]],[.,[.,.]]] [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
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.