Identifier

Mp00033:
Dyck paths

Mp00081: Standard tableaux

Mp00071: Permutations

**—**to two-row standard tableau⟶ Standard tableauxMp00081: Standard tableaux

**—**reading word permutation⟶ PermutationsMp00071: Permutations

**—**descent composition⟶ Integer compositions
Images

=>

Cc0005;cc-rep-0Cc0007;cc-rep-1

[1,0]=>[[1],[2]]=>[2,1]=>[1,1]
[1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>[2,2]
[1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,2]
[1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[3,3]
[1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[3,3]
[1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[3,3]
[1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[3,3]
[1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[3,3]
[1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>[4,4]
[1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>[2,4,7,8,1,3,5,6]=>[4,4]
[1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[2,5,6,8,1,3,4,7]=>[4,4]
[1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>[2,5,7,8,1,3,4,6]=>[4,4]
[1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[2,6,7,8,1,3,4,5]=>[4,4]
[1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[3,4,6,8,1,2,5,7]=>[4,4]
[1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>[3,4,7,8,1,2,5,6]=>[4,4]
[1,1,0,1,0,0,1,0]=>[[1,2,4,7],[3,5,6,8]]=>[3,5,6,8,1,2,4,7]=>[4,4]
[1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[3,5,7,8,1,2,4,6]=>[4,4]
[1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[3,6,7,8,1,2,4,5]=>[4,4]
[1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[4,5,6,8,1,2,3,7]=>[4,4]
[1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>[4,5,7,8,1,2,3,6]=>[4,4]
[1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[4,6,7,8,1,2,3,5]=>[4,4]
[1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[4,4]
[1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9],[2,4,6,8,10]]=>[2,4,6,8,10,1,3,5,7,9]=>[5,5]
[1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9,11],[2,4,6,8,10,12]]=>[2,4,6,8,10,12,1,3,5,7,9,11]=>[6,6]

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

**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 composition**

Description

The descent composition of a permutation.

The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.

The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.

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