Identifier

Mp00033:
Dyck paths

Mp00084: Standard tableaux

Mp00081: Standard tableaux

Mp00061: Permutations

**—**to two-row standard tableau⟶ Standard tableauxMp00084: Standard tableaux

**—**conjugate⟶ Standard tableauxMp00081: Standard tableaux

**—**reading word permutation⟶ PermutationsMp00061: Permutations

**—**to increasing tree⟶ Binary trees
Images

=>

Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0007;cc-rep-2Cc0010;cc-rep-4

[1,0]=>[[1],[2]]=>[[1,2]]=>[1,2]=>[.,[.,.]]
[1,0,1,0]=>[[1,3],[2,4]]=>[[1,2],[3,4]]=>[3,4,1,2]=>[[.,[.,.]],[.,.]]
[1,1,0,0]=>[[1,2],[3,4]]=>[[1,3],[2,4]]=>[2,4,1,3]=>[[.,[.,.]],[.,.]]
[1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[[[.,[.,.]],[.,.]],[.,.]]
[1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[[1,2],[3,5],[4,6]]=>[4,6,3,5,1,2]=>[[[.,[.,.]],[.,.]],[.,.]]
[1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[[1,3],[2,4],[5,6]]=>[5,6,2,4,1,3]=>[[[.,[.,.]],[.,.]],[.,.]]
[1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[[1,3],[2,5],[4,6]]=>[4,6,2,5,1,3]=>[[[.,[.,.]],[.,.]],[.,.]]
[1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[[1,4],[2,5],[3,6]]=>[3,6,2,5,1,4]=>[[[.,[.,.]],[.,.]],[.,.]]
[1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]

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.

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

**conjugate**

Description

Sends a standard tableau to its conjugate tableau.

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.

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.

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