Identifier

Mp00033:
Dyck paths

Mp00081: Standard tableaux

Mp00109: Permutations

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

**—**reading word permutation⟶ PermutationsMp00109: Permutations

**—**descent word⟶ Binary words
Images

=>

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

[1,0]=>[[1],[2]]=>[2,1]=>1
[1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>010
[1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>010
[1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>00100
[1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>00100
[1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>00100
[1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>00100
[1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>00100
[1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>0001000
[1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[2,6,7,8,1,3,4,5]=>0001000
[1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[3,5,7,8,1,2,4,6]=>0001000
[1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[3,6,7,8,1,2,4,5]=>0001000
[1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[4,5,6,8,1,2,3,7]=>0001000
[1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[4,6,7,8,1,2,3,5]=>0001000
[1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>0001000
[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]=>000010000
[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]=>00000100000

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 word**

Description

The descent positions of a permutation as a binary word.

For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.

Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.

For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.

Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.

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