Identifier
Mp00058:
Perfect matchings
—to permutation⟶
Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00239: Permutations —Corteel⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Images
[(1,2)] => [2,1] => [2,1] => [2]
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [2,2]
[(1,3),(2,4)] => [3,4,1,2] => [4,3,2,1] => [1,2,1]
[(1,4),(2,3)] => [4,3,2,1] => [3,4,1,2] => [4]
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,2,2]
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,3,2,1,6,5] => [1,2,1,2]
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [3,4,1,2,6,5] => [4,2]
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [3,6,1,5,4,2] => [3,2,1]
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [3,5,1,6,2,4] => [3,3]
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [4,6,5,2,1,3] => [2,2,2]
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [5,6,4,2,3,1] => [2,3,1]
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [6,5,4,3,2,1] => [1,1,2,1,1]
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [6,3,2,5,4,1] => [1,2,2,1]
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,5,4,3] => [2,1,2,1]
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,5,6,3,4] => [2,4]
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [5,3,2,6,1,4] => [1,2,3]
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [6,4,5,3,1,2] => [1,3,2]
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [5,4,6,1,3,2] => [1,4,1]
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [4,5,6,1,2,3] => [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] => [2,2,2,2]
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [4,3,2,1,6,5,8,7] => [1,2,1,2,2]
[(1,4),(2,3),(5,6),(7,8)] => [4,3,2,1,6,5,8,7] => [3,4,1,2,6,5,8,7] => [4,2,2]
[(1,5),(2,3),(4,6),(7,8)] => [5,3,2,6,1,4,8,7] => [3,6,1,5,4,2,8,7] => [3,2,1,2]
[(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [3,5,1,6,2,4,8,7] => [3,3,2]
[(1,7),(2,3),(4,5),(6,8)] => [7,3,2,5,4,8,1,6] => [3,5,1,8,2,7,6,4] => [3,2,2,1]
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [3,5,1,7,2,8,4,6] => [3,2,3]
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [4,7,5,2,1,8,3,6] => [2,2,1,3]
[(1,7),(2,4),(3,5),(6,8)] => [7,4,5,2,3,8,1,6] => [4,8,5,2,1,7,6,3] => [2,2,1,2,1]
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [4,6,5,2,1,3,8,7] => [2,2,2,2]
[(1,5),(2,4),(3,6),(7,8)] => [5,4,6,2,1,3,8,7] => [5,6,4,2,3,1,8,7] => [2,3,1,2]
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [6,5,4,3,2,1,8,7] => [1,1,2,1,1,2]
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [6,3,2,5,4,1,8,7] => [1,2,2,1,2]
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,6,5,4,3,8,7] => [2,1,2,1,2]
[(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [2,1,5,6,3,4,8,7] => [2,4,2]
[(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [5,3,2,6,1,4,8,7] => [1,2,3,2]
[(1,4),(2,6),(3,5),(7,8)] => [4,6,5,1,3,2,8,7] => [6,4,5,3,1,2,8,7] => [1,3,2,2]
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => [5,4,6,1,3,2,8,7] => [1,4,1,2]
[(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [4,5,6,1,2,3,8,7] => [6,2]
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [4,5,8,1,2,7,6,3] => [5,2,1]
[(1,8),(2,5),(3,4),(6,7)] => [8,5,4,3,2,7,6,1] => [4,5,7,1,2,8,3,6] => [5,3]
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [4,6,8,1,7,3,2,5] => [4,2,2]
[(1,7),(2,6),(3,4),(5,8)] => [7,6,4,3,8,2,1,5] => [4,7,8,1,6,3,5,2] => [4,3,1]
[(1,6),(2,7),(3,4),(5,8)] => [6,7,4,3,8,1,2,5] => [7,4,8,1,6,5,3,2] => [1,3,2,1,1]
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [5,4,8,1,3,7,6,2] => [1,4,2,1]
[(1,4),(2,7),(3,5),(6,8)] => [4,7,5,1,3,8,2,6] => [8,4,5,3,1,7,6,2] => [1,3,1,2,1]
[(1,3),(2,7),(4,5),(6,8)] => [3,7,1,5,4,8,2,6] => [5,3,2,8,1,7,6,4] => [1,2,2,2,1]
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [2,1,5,8,3,7,6,4] => [2,3,2,1]
[(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [2,1,5,7,3,8,4,6] => [2,3,3]
[(1,3),(2,8),(4,5),(6,7)] => [3,8,1,5,4,7,6,2] => [5,3,2,7,1,8,4,6] => [1,2,2,3]
[(1,4),(2,8),(3,5),(6,7)] => [4,8,5,1,3,7,6,2] => [7,4,5,3,1,8,2,6] => [1,3,1,3]
[(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => [5,4,7,1,3,8,2,6] => [1,4,3]
[(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => [6,4,8,1,7,5,2,3] => [1,3,2,2]
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [6,4,7,1,8,2,5,3] => [1,3,3,1]
[(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [4,6,7,1,8,2,3,5] => [4,4]
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [5,6,8,7,2,1,3,4] => [3,2,3]
[(1,7),(2,8),(3,5),(4,6)] => [7,8,5,6,3,4,1,2] => [6,5,8,7,2,1,4,3] => [1,2,2,2,1]
[(1,6),(2,8),(3,5),(4,7)] => [6,8,5,7,3,1,4,2] => [7,5,8,6,2,4,1,3] => [1,2,3,2]
[(1,5),(2,8),(3,6),(4,7)] => [5,8,6,7,1,3,4,2] => [8,5,7,6,4,2,1,3] => [1,2,2,1,2]
[(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => [8,4,6,3,7,2,1,5] => [1,3,2,2]
[(1,3),(2,8),(4,6),(5,7)] => [3,8,1,6,7,4,5,2] => [6,3,2,8,7,4,1,5] => [1,2,1,2,2]
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [2,1,6,8,7,4,3,5] => [2,2,2,2]
[(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [2,1,7,8,6,4,5,3] => [2,2,3,1]
[(1,3),(2,7),(4,6),(5,8)] => [3,7,1,6,8,4,2,5] => [7,3,2,8,6,4,5,1] => [1,2,1,3,1]
[(1,4),(2,7),(3,6),(5,8)] => [4,7,6,1,8,3,2,5] => [8,4,7,3,6,2,5,1] => [1,3,3,1]
[(1,5),(2,7),(3,6),(4,8)] => [5,7,6,8,1,3,2,4] => [8,6,7,5,4,2,3,1] => [1,2,2,2,1]
[(1,6),(2,7),(3,5),(4,8)] => [6,7,5,8,3,1,2,4] => [7,6,8,5,2,4,3,1] => [1,2,3,1,1]
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [6,7,8,5,2,3,4,1] => [3,4,1]
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [5,7,8,6,2,3,1,4] => [3,3,2]
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [5,8,7,6,3,2,1,4] => [2,1,2,1,2]
[(1,7),(2,5),(3,6),(4,8)] => [7,5,6,8,2,3,1,4] => [6,8,7,5,3,2,4,1] => [2,1,2,2,1]
[(1,6),(2,5),(3,7),(4,8)] => [6,5,7,8,2,1,3,4] => [7,8,6,5,3,4,2,1] => [2,1,3,1,1]
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [8,7,6,5,4,3,2,1] => [1,1,1,2,1,1,1]
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [8,7,4,3,6,5,2,1] => [1,1,2,2,1,1]
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [8,3,2,7,6,5,4,1] => [1,2,1,2,1,1]
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => [2,1,1,2,1,1]
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,8,5,4,7,6,3] => [2,1,2,2,1]
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [8,3,2,5,4,7,6,1] => [1,2,2,2,1]
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [8,5,4,3,2,7,6,1] => [1,1,2,1,2,1]
[(1,5),(2,4),(3,7),(6,8)] => [5,4,7,2,1,8,3,6] => [5,8,4,2,3,7,6,1] => [2,3,2,1]
[(1,6),(2,4),(3,7),(5,8)] => [6,4,7,2,8,1,3,5] => [7,8,4,2,6,5,3,1] => [2,2,2,1,1]
[(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => [4,8,7,2,6,3,5,1] => [2,2,3,1]
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [4,8,6,2,7,3,1,5] => [2,2,2,2]
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [3,6,1,8,7,4,2,5] => [3,1,2,2]
[(1,7),(2,3),(4,6),(5,8)] => [7,3,2,6,8,4,1,5] => [3,7,1,8,6,4,5,2] => [3,1,3,1]
[(1,6),(2,3),(4,7),(5,8)] => [6,3,2,7,8,1,4,5] => [3,8,1,7,6,5,4,2] => [3,1,2,1,1]
[(1,5),(2,3),(4,7),(6,8)] => [5,3,2,7,1,8,4,6] => [3,8,1,5,4,7,6,2] => [3,2,2,1]
[(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [3,4,1,2,8,7,6,5] => [4,1,2,1]
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [4,3,2,1,8,7,6,5] => [1,2,1,1,2,1]
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,4,3,8,7,6,5] => [2,2,1,2,1]
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [2,1,4,3,7,8,5,6] => [2,2,4]
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [4,3,2,1,7,8,5,6] => [1,2,1,4]
[(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [3,4,1,2,7,8,5,6] => [4,4]
[(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [3,7,1,5,4,8,2,6] => [3,2,3]
[(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [3,8,1,6,7,5,2,4] => [3,3,2]
[(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => [3,7,1,6,8,2,5,4] => [3,4,1]
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [3,6,1,7,8,2,4,5] => [3,5]
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [4,7,6,2,8,1,3,5] => [2,2,4]
[(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => [6,7,4,2,8,1,5,3] => [2,2,3,1]
[(1,6),(2,4),(3,8),(5,7)] => [6,4,8,2,7,1,5,3] => [6,8,4,2,7,5,1,3] => [2,2,2,2]
[(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [5,7,4,2,3,8,1,6] => [2,3,3]
[(1,4),(2,5),(3,8),(6,7)] => [4,5,8,1,2,7,6,3] => [7,5,4,3,2,8,1,6] => [1,1,2,1,3]
>>> Load all 303 entries. <<<Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
Map
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
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