Identifier

Mp00033:
Dyck paths

Mp00081: Standard tableaux

Mp00087: Permutations

Mp00130: Permutations

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

**—**reading word permutation⟶ PermutationsMp00087: Permutations

**—**inverse first fundamental transformation⟶ PermutationsMp00130: Permutations

**—**descent tops⟶ Binary words
Images

=>

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

[1,0]=>[[1],[2]]=>[2,1]=>[2,1]=>1
[1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>[4,3,1,2]=>011
[1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>[3,1,4,2]=>011
[1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[4,1,2,6,5,3]=>00111
[1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[6,4,1,2,5,3]=>00111
[1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[6,5,2,4,1,3]=>00111
[1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[5,2,6,4,1,3]=>00111
[1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[4,1,5,2,6,3]=>00111
[1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>[6,3,8,7,5,1,2,4]=>0001111
[1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>[2,4,7,8,1,3,5,6]=>[8,6,3,7,5,1,2,4]=>0001111
[1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[2,5,6,8,1,3,4,7]=>[5,1,2,6,3,8,7,4]=>0001111
[1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>[2,5,7,8,1,3,4,6]=>[5,1,2,8,6,3,7,4]=>0001111
[1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[3,4,6,8,1,2,5,7]=>[8,7,5,1,3,6,2,4]=>0001111
[1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[4,5,6,8,1,2,3,7]=>[8,7,3,6,2,5,1,4]=>0001111
[1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>[4,5,7,8,1,2,3,6]=>[7,3,8,6,2,5,1,4]=>0001111
[1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[4,6,7,8,1,2,3,5]=>[6,2,7,3,8,5,1,4]=>0001111
[1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[5,1,6,2,7,3,8,4]=>0001111

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

**inverse first fundamental transformation**

Description

Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.

Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.

In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.

In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

Map

**descent tops**

Description

The descent tops of a permutation as a binary word.

Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.

Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.

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