Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Images
[(1,2)] => [2,1] => [1,2] => [2]
[(1,2),(3,4)] => [2,1,4,3] => [3,2,1,4] => [2,2]
[(1,3),(2,4)] => [3,4,1,2] => [4,1,2,3] => [4]
[(1,4),(2,3)] => [3,4,2,1] => [1,4,2,3] => [1,3]
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [3,2,5,4,1,6] => [2,2,2]
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,5,2,3,1,6] => [4,2]
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [5,4,2,3,1,6] => [1,3,2]
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [6,4,2,1,3,5] => [1,2,3]
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [1,4,2,6,3,5] => [1,2,3]
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [1,5,6,2,3,4] => [1,5]
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [6,5,1,2,3,4] => [1,5]
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [5,6,1,2,3,4] => [6]
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [4,6,2,1,3,5] => [3,3]
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [3,2,6,1,4,5] => [2,4]
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [3,2,1,6,4,5] => [2,1,3]
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [4,1,2,6,3,5] => [3,3]
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [5,1,6,2,3,4] => [2,4]
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6,1,5,2,3,4] => [2,4]
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [1,6,5,2,3,4] => [1,1,4]
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => [2,2,2,2]
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [4,5,2,3,7,6,1,8] => [4,2,2]
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [5,4,2,3,7,6,1,8] => [1,3,2,2]
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [6,4,2,7,3,5,1,8] => [1,2,3,2]
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [7,4,2,6,3,5,1,8] => [1,2,3,2]
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [8,4,2,6,3,1,5,7] => [1,2,2,3]
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [1,4,2,6,3,8,5,7] => [1,2,2,3]
[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [1,5,6,2,3,8,4,7] => [1,4,3]
[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [8,5,6,2,3,1,4,7] => [1,4,3]
[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [7,5,6,2,3,4,1,8] => [1,5,2]
[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [6,5,7,2,3,4,1,8] => [1,5,2]
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [5,6,7,2,3,4,1,8] => [6,2]
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [4,6,2,7,3,5,1,8] => [3,3,2]
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [3,2,6,7,4,5,1,8] => [2,4,2]
[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [3,2,7,6,4,5,1,8] => [2,1,3,2]
[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [4,7,2,6,3,5,1,8] => [3,3,2]
[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [5,7,6,2,3,4,1,8] => [2,4,2]
[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [6,7,5,2,3,4,1,8] => [2,4,2]
[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [7,6,5,2,3,4,1,8] => [1,1,4,2]
[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [8,6,5,2,3,1,4,7] => [1,1,3,3]
[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [1,6,5,2,3,8,4,7] => [1,1,3,3]
[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [1,7,5,2,8,3,4,6] => [1,1,2,4]
[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [8,7,5,2,1,3,4,6] => [1,1,2,4]
[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => [7,8,5,2,1,3,4,6] => [2,2,4]
[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [6,8,5,2,3,1,4,7] => [2,3,3]
[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [5,8,6,2,3,1,4,7] => [2,3,3]
[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [4,8,2,6,3,1,5,7] => [3,2,3]
[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [3,2,8,6,4,1,5,7] => [2,1,2,3]
[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [3,2,1,6,4,8,5,7] => [2,1,2,3]
[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [4,1,2,6,3,8,5,7] => [3,2,3]
[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [5,1,6,2,3,8,4,7] => [2,3,3]
[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [6,1,5,2,3,8,4,7] => [2,3,3]
[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [7,1,5,2,8,3,4,6] => [2,2,4]
[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [8,1,5,2,7,3,4,6] => [2,2,4]
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [1,8,5,2,7,3,4,6] => [1,1,2,4]
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [1,8,6,7,2,3,4,5] => [1,1,6]
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [8,1,6,7,2,3,4,5] => [2,6]
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [7,1,6,8,2,3,4,5] => [2,6]
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [6,1,7,8,2,3,4,5] => [2,6]
[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [5,1,7,2,8,3,4,6] => [2,2,4]
[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [4,1,2,7,8,3,5,6] => [3,5]
[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [3,2,1,7,8,4,5,6] => [2,1,5]
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [3,2,8,7,1,4,5,6] => [2,1,5]
[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [4,8,2,7,1,3,5,6] => [3,5]
[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [5,8,7,2,1,3,4,6] => [2,2,4]
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [6,8,7,1,2,3,4,5] => [2,6]
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [7,8,6,1,2,3,4,5] => [2,6]
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [8,7,6,1,2,3,4,5] => [1,1,6]
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [1,7,6,8,2,3,4,5] => [1,1,6]
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [1,6,7,8,2,3,4,5] => [1,7]
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [8,6,7,1,2,3,4,5] => [1,7]
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [7,6,8,1,2,3,4,5] => [1,7]
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [6,7,8,1,2,3,4,5] => [8]
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [5,7,8,2,1,3,4,6] => [4,4]
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [4,7,2,8,1,3,5,6] => [3,5]
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [3,2,7,8,1,4,5,6] => [2,6]
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [3,2,6,8,4,1,5,7] => [2,3,3]
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [4,6,2,8,3,1,5,7] => [3,2,3]
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [5,6,8,2,3,1,4,7] => [5,3]
[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [6,5,8,2,3,1,4,7] => [1,4,3]
[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [7,5,8,2,1,3,4,6] => [1,3,4]
[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [8,5,7,2,1,3,4,6] => [1,3,4]
[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [1,5,7,2,8,3,4,6] => [1,3,4]
[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [1,4,2,7,8,3,5,6] => [1,2,5]
[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [8,4,2,7,1,3,5,6] => [1,2,5]
[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [7,4,2,8,1,3,5,6] => [1,2,5]
[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [6,4,2,8,3,1,5,7] => [1,2,2,3]
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [5,4,2,3,8,1,6,7] => [1,3,4]
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [4,5,2,3,8,1,6,7] => [4,4]
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [3,2,5,4,8,1,6,7] => [2,2,4]
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [3,2,5,4,1,8,6,7] => [2,2,1,3]
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [4,5,2,3,1,8,6,7] => [4,1,3]
[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [5,4,2,3,1,8,6,7] => [1,3,1,3]
[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [6,4,2,1,3,8,5,7] => [1,2,2,3]
[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => [7,4,2,1,8,3,5,6] => [1,2,1,4]
[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => [8,4,2,1,7,3,5,6] => [1,2,1,4]
[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [1,4,2,8,7,3,5,6] => [1,2,1,4]
[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [1,5,8,2,7,3,4,6] => [1,3,4]
[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [8,5,1,2,7,3,4,6] => [1,3,4]
[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [7,5,1,2,8,3,4,6] => [1,3,4]
[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [6,5,1,2,3,8,4,7] => [1,4,3]
[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [5,6,1,2,3,8,4,7] => [5,3]
>>> Load all 126 entries. <<<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
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.
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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