Identifier

Mp00146:
Dyck paths

Mp00058: Perfect matchings

Mp00087: Permutations

Mp00238: Permutations

**—**to tunnel matching⟶ Perfect matchingsMp00058: Perfect matchings

**—**to permutation⟶ PermutationsMp00087: Permutations

**—**inverse first fundamental transformation⟶ PermutationsMp00238: Permutations

**—**Clarke-Steingrimsson-Zeng⟶ Permutations
Images

=>

Cc0005;cc-rep-0Cc0012;cc-rep-1

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

Map

**to tunnel matching**

Description

Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.

This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.

This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.

Map

**to permutation**

Description

Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.

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

**Clarke-Steingrimsson-Zeng**

Description

The Clarke-Steingrimsson-Zeng map sending descents to excedances.

This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies

$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$

where

This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies

$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$

where

- $des$ is the number of descents, St000021The number of descents of a permutation.,
- $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
- $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
- $Ebot$ is the sum of the excedance bottoms,
- $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
- $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
- $Res$ is the sum of the (right) embracing numbers,
- $Ine$ is the sum of the side numbers.

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