Identifier

Mp00058:
Perfect matchings

Mp00087: Permutations

Mp00069: Permutations

Mp00061: Permutations

**—**to permutation⟶ PermutationsMp00087: Permutations

**—**inverse first fundamental transformation⟶ PermutationsMp00069: Permutations

**—**complement⟶ PermutationsMp00061: Permutations

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

=>

Cc0012;cc-rep-0Cc0010;cc-rep-4

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

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

**complement**

Description

Sents a permutation to its complement.

The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

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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