Identifier
Mp00116:
Perfect matchings
—Kasraoui-Zeng⟶
Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Images
[(1,2)] => [(1,2)] => [2,1] => [2]
[(1,2),(3,4)] => [(1,2),(3,4)] => [2,1,4,3] => [2,2]
[(1,3),(2,4)] => [(1,4),(2,3)] => [3,4,2,1] => [3,1]
[(1,4),(2,3)] => [(1,3),(2,4)] => [3,4,1,2] => [4]
[(1,2),(3,4),(5,6)] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,2,2]
[(1,3),(2,4),(5,6)] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [3,1,2]
[(1,4),(2,3),(5,6)] => [(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,2]
[(1,5),(2,3),(4,6)] => [(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [3,2,1]
[(1,6),(2,3),(4,5)] => [(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,3]
[(1,6),(2,4),(3,5)] => [(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [4,2]
[(1,5),(2,4),(3,6)] => [(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [5,1]
[(1,4),(2,5),(3,6)] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [4,1,1]
[(1,3),(2,5),(4,6)] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [3,2,1]
[(1,2),(3,5),(4,6)] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,3,1]
[(1,2),(3,6),(4,5)] => [(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,4]
[(1,3),(2,6),(4,5)] => [(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [3,3]
[(1,4),(2,6),(3,5)] => [(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [4,2]
[(1,5),(2,6),(3,4)] => [(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [5,1]
[(1,6),(2,5),(3,4)] => [(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [6]
[(1,2),(3,4),(5,6),(7,8)] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,2,2,2]
[(1,3),(2,4),(5,6),(7,8)] => [(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [3,1,2,2]
[(1,4),(2,3),(5,6),(7,8)] => [(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [4,2,2]
[(1,5),(2,3),(4,6),(7,8)] => [(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [3,2,1,2]
[(1,6),(2,3),(4,5),(7,8)] => [(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [3,3,2]
[(1,7),(2,3),(4,5),(6,8)] => [(1,3),(2,5),(4,8),(6,7)] => [3,5,1,7,2,8,6,4] => [3,2,2,1]
[(1,8),(2,3),(4,5),(6,7)] => [(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [3,2,3]
[(1,8),(2,4),(3,5),(6,7)] => [(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [4,1,3]
[(1,7),(2,4),(3,5),(6,8)] => [(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [4,1,2,1]
[(1,6),(2,4),(3,5),(7,8)] => [(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [4,2,2]
[(1,5),(2,4),(3,6),(7,8)] => [(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [5,1,2]
[(1,4),(2,5),(3,6),(7,8)] => [(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [4,1,1,2]
[(1,3),(2,5),(4,6),(7,8)] => [(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [3,2,1,2]
[(1,2),(3,5),(4,6),(7,8)] => [(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [2,3,1,2]
[(1,2),(3,6),(4,5),(7,8)] => [(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,4,2]
[(1,3),(2,6),(4,5),(7,8)] => [(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [3,3,2]
[(1,4),(2,6),(3,5),(7,8)] => [(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [4,2,2]
[(1,5),(2,6),(3,4),(7,8)] => [(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [5,1,2]
[(1,6),(2,5),(3,4),(7,8)] => [(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [6,2]
[(1,7),(2,5),(3,4),(6,8)] => [(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [5,2,1]
[(1,8),(2,5),(3,4),(6,7)] => [(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [5,3]
[(1,8),(2,6),(3,4),(5,7)] => [(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [4,2,2]
[(1,7),(2,6),(3,4),(5,8)] => [(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [4,3,1]
[(1,6),(2,7),(3,4),(5,8)] => [(1,4),(2,8),(3,7),(5,6)] => [4,6,7,1,8,5,3,2] => [4,2,1,1]
[(1,5),(2,7),(3,4),(6,8)] => [(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [5,2,1]
[(1,4),(2,7),(3,5),(6,8)] => [(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [4,1,2,1]
[(1,3),(2,7),(4,5),(6,8)] => [(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [3,2,2,1]
[(1,2),(3,7),(4,5),(6,8)] => [(1,2),(3,5),(4,8),(6,7)] => [2,1,5,7,3,8,6,4] => [2,3,2,1]
[(1,2),(3,8),(4,5),(6,7)] => [(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,3,3]
[(1,3),(2,8),(4,5),(6,7)] => [(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [3,2,3]
[(1,4),(2,8),(3,5),(6,7)] => [(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [4,1,3]
[(1,5),(2,8),(3,4),(6,7)] => [(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [5,3]
[(1,6),(2,8),(3,4),(5,7)] => [(1,4),(2,7),(3,8),(5,6)] => [4,6,7,1,8,5,2,3] => [4,2,2]
[(1,7),(2,8),(3,4),(5,6)] => [(1,4),(2,6),(3,8),(5,7)] => [4,6,7,1,8,2,5,3] => [4,3,1]
[(1,8),(2,7),(3,4),(5,6)] => [(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [4,4]
[(1,8),(2,7),(3,5),(4,6)] => [(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [5,3]
[(1,7),(2,8),(3,5),(4,6)] => [(1,6),(2,5),(3,8),(4,7)] => [5,6,7,8,2,1,4,3] => [5,2,1]
[(1,6),(2,8),(3,5),(4,7)] => [(1,7),(2,5),(3,8),(4,6)] => [5,6,7,8,2,4,1,3] => [6,2]
[(1,5),(2,8),(3,6),(4,7)] => [(1,7),(2,6),(3,8),(4,5)] => [5,6,7,8,4,2,1,3] => [5,1,2]
[(1,4),(2,8),(3,6),(5,7)] => [(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [4,2,2]
[(1,3),(2,8),(4,6),(5,7)] => [(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [3,3,2]
[(1,2),(3,8),(4,6),(5,7)] => [(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [2,4,2]
[(1,2),(3,7),(4,6),(5,8)] => [(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [2,5,1]
[(1,3),(2,7),(4,6),(5,8)] => [(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [3,4,1]
[(1,4),(2,7),(3,6),(5,8)] => [(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [4,3,1]
[(1,5),(2,7),(3,6),(4,8)] => [(1,8),(2,6),(3,7),(4,5)] => [5,6,7,8,4,2,3,1] => [5,2,1]
[(1,6),(2,7),(3,5),(4,8)] => [(1,8),(2,5),(3,7),(4,6)] => [5,6,7,8,2,4,3,1] => [6,1,1]
[(1,7),(2,6),(3,5),(4,8)] => [(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [7,1]
[(1,8),(2,6),(3,5),(4,7)] => [(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [6,2]
[(1,8),(2,5),(3,6),(4,7)] => [(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [5,1,2]
[(1,7),(2,5),(3,6),(4,8)] => [(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [5,2,1]
[(1,6),(2,5),(3,7),(4,8)] => [(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [6,1,1]
[(1,5),(2,6),(3,7),(4,8)] => [(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [5,1,1,1]
[(1,4),(2,6),(3,7),(5,8)] => [(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [4,2,1,1]
[(1,3),(2,6),(4,7),(5,8)] => [(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [3,3,1,1]
[(1,2),(3,6),(4,7),(5,8)] => [(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => [2,4,1,1]
[(1,2),(3,5),(4,7),(6,8)] => [(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [2,3,2,1]
[(1,3),(2,5),(4,7),(6,8)] => [(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [3,2,2,1]
[(1,4),(2,5),(3,7),(6,8)] => [(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [4,1,2,1]
[(1,5),(2,4),(3,7),(6,8)] => [(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [5,2,1]
[(1,6),(2,4),(3,7),(5,8)] => [(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [4,2,1,1]
[(1,7),(2,4),(3,6),(5,8)] => [(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [4,3,1]
[(1,8),(2,4),(3,6),(5,7)] => [(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [4,2,2]
[(1,8),(2,3),(4,6),(5,7)] => [(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [3,3,2]
[(1,7),(2,3),(4,6),(5,8)] => [(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [3,4,1]
[(1,6),(2,3),(4,7),(5,8)] => [(1,3),(2,8),(4,7),(5,6)] => [3,6,1,7,8,5,4,2] => [3,3,1,1]
[(1,5),(2,3),(4,7),(6,8)] => [(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [3,2,2,1]
[(1,4),(2,3),(5,7),(6,8)] => [(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [4,3,1]
[(1,3),(2,4),(5,7),(6,8)] => [(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [3,1,3,1]
[(1,2),(3,4),(5,7),(6,8)] => [(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [2,2,3,1]
[(1,2),(3,4),(5,8),(6,7)] => [(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,2,4]
[(1,3),(2,4),(5,8),(6,7)] => [(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [3,1,4]
[(1,4),(2,3),(5,8),(6,7)] => [(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [4,4]
[(1,5),(2,3),(4,8),(6,7)] => [(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [3,2,3]
[(1,6),(2,3),(4,8),(5,7)] => [(1,3),(2,7),(4,8),(5,6)] => [3,6,1,7,8,5,2,4] => [3,3,2]
[(1,7),(2,3),(4,8),(5,6)] => [(1,3),(2,6),(4,8),(5,7)] => [3,6,1,7,8,2,5,4] => [3,4,1]
[(1,8),(2,3),(4,7),(5,6)] => [(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [3,5]
[(1,8),(2,4),(3,7),(5,6)] => [(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [4,4]
[(1,7),(2,4),(3,8),(5,6)] => [(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [4,3,1]
[(1,6),(2,4),(3,8),(5,7)] => [(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [4,2,2]
[(1,5),(2,4),(3,8),(6,7)] => [(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [5,3]
[(1,4),(2,5),(3,8),(6,7)] => [(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [4,1,3]
>>> Load all 170 entries. <<<Map
Kasraoui-Zeng
Description
The Kasraoui-Zeng involution for perfect matchings.
This yields the perfect matching with the number of nestings and crossings exchanged.
This yields the perfect matching with the number of nestings and crossings exchanged.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
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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