Identifier
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00066: Permutations —inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Images
[1,0] => [2,1] => [2,1] => [1,2] => [2]
[1,0,1,0] => [3,1,2] => [2,3,1] => [1,2,3] => [3]
[1,1,0,0] => [2,3,1] => [3,1,2] => [3,1,2] => [3]
[1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => [1,2,3,4] => [4]
[1,0,1,1,0,0] => [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [2,2]
[1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => [3,1,2,4] => [4]
[1,1,0,1,0,0] => [4,3,1,2] => [3,4,2,1] => [1,4,2,3] => [1,3]
[1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => [3,4,1,2] => [4]
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => [1,2,3,4,5] => [5]
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,3,5,1,4] => [5,2,3,1,4] => [3,2]
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [2,4,1,5,3] => [4,2,1,3,5] => [2,3]
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [2,4,5,3,1] => [1,2,5,3,4] => [2,3]
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [2,5,1,3,4] => [4,2,5,1,3] => [2,3]
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,1,4,5,2] => [3,1,2,4,5] => [5]
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,1,5,2,4] => [3,5,2,1,4] => [3,2]
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,4,2,5,1] => [1,4,2,3,5] => [1,4]
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [3,4,5,2,1] => [1,5,2,3,4] => [1,4]
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,5,2,1,4] => [5,4,2,1,3] => [1,2,2]
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,1,2,5,3] => [3,4,1,2,5] => [5]
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,1,5,3,2] => [3,1,5,2,4] => [2,3]
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,5,2,3,1] => [1,4,5,2,3] => [1,4]
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => [3,4,5,1,2] => [5]
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [6]
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => [4,2]
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [2,3,5,1,6,4] => [5,2,3,1,4,6] => [3,3]
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [2,3,5,6,4,1] => [1,2,3,6,4,5] => [3,3]
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [2,3,6,1,4,5] => [5,2,3,6,1,4] => [3,3]
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [2,4,1,5,6,3] => [4,2,1,3,5,6] => [2,4]
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [2,4,1,6,3,5] => [4,2,6,3,1,5] => [2,2,2]
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [2,4,5,3,6,1] => [1,2,5,3,4,6] => [2,4]
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [2,4,5,6,3,1] => [1,2,6,3,4,5] => [2,4]
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [2,4,6,3,1,5] => [6,2,5,3,1,4] => [2,2,2]
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [2,5,1,3,6,4] => [4,2,5,1,3,6] => [2,4]
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [2,5,1,6,4,3] => [4,2,1,6,3,5] => [2,1,3]
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [2,5,6,3,4,1] => [1,2,5,6,3,4] => [2,4]
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [2,6,1,3,4,5] => [4,2,5,6,1,3] => [2,4]
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => [3,1,2,4,5,6] => [6]
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,1,4,6,2,5] => [3,6,2,4,1,5] => [4,2]
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [3,1,5,2,6,4] => [3,5,2,1,4,6] => [3,3]
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [3,1,5,6,4,2] => [3,1,2,6,4,5] => [3,3]
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,1,6,2,4,5] => [3,5,2,6,1,4] => [3,3]
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,4,2,5,6,1] => [1,4,2,3,5,6] => [1,5]
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,4,2,6,1,5] => [6,4,2,3,1,5] => [1,3,2]
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [3,4,5,2,6,1] => [1,5,2,3,4,6] => [1,5]
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [3,4,5,6,1,2] => [6,1,2,3,4,5] => [6]
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [3,4,6,2,1,5] => [6,5,2,3,1,4] => [1,3,2]
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,5,2,1,6,4] => [5,4,2,1,3,6] => [1,2,3]
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,5,2,6,4,1] => [1,4,2,6,3,5] => [1,2,3]
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [3,5,6,2,4,1] => [1,5,2,6,3,4] => [1,2,3]
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,6,2,1,4,5] => [5,4,2,6,1,3] => [1,2,3]
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [4,1,2,5,6,3] => [3,4,1,2,5,6] => [6]
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [4,1,2,6,3,5] => [3,4,6,2,1,5] => [4,2]
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,1,5,3,6,2] => [3,1,5,2,4,6] => [2,4]
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [4,1,5,6,3,2] => [3,1,6,2,4,5] => [2,4]
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,1,6,3,2,5] => [3,6,5,2,1,4] => [2,2,2]
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,5,2,3,6,1] => [1,4,5,2,3,6] => [1,5]
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [4,5,2,6,3,1] => [1,4,6,2,3,5] => [1,5]
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => [1,6,5,2,3,4] => [1,1,4]
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,6,2,3,1,5] => [6,4,5,2,1,3] => [1,3,2]
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,1,2,3,6,4] => [3,4,5,1,2,6] => [6]
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,1,2,6,4,3] => [3,4,1,6,2,5] => [3,3]
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,1,6,3,4,2] => [3,1,5,6,2,4] => [2,4]
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,6,2,3,4,1] => [1,4,5,6,2,3] => [1,5]
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [3,4,5,6,1,2] => [6]
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [7]
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => [5,2]
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [2,3,4,6,1,7,5] => [6,2,3,4,1,5,7] => [4,3]
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [2,3,4,6,7,5,1] => [1,2,3,4,7,5,6] => [4,3]
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [2,3,4,7,1,5,6] => [6,2,3,4,7,1,5] => [4,3]
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [2,3,5,1,6,7,4] => [5,2,3,1,4,6,7] => [3,4]
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [2,3,5,1,7,4,6] => [5,2,3,7,4,1,6] => [3,2,2]
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [2,3,5,6,4,7,1] => [1,2,3,6,4,5,7] => [3,4]
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [2,3,5,6,7,4,1] => [1,2,3,7,4,5,6] => [3,4]
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [2,3,5,7,4,1,6] => [7,2,3,6,4,1,5] => [3,2,2]
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [2,3,6,1,4,7,5] => [5,2,3,6,1,4,7] => [3,4]
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [2,3,6,1,7,5,4] => [5,2,3,1,7,4,6] => [3,1,3]
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [2,3,6,7,4,5,1] => [1,2,3,6,7,4,5] => [3,4]
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [2,3,7,1,4,5,6] => [5,2,3,6,7,1,4] => [3,4]
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [2,4,1,5,6,7,3] => [4,2,1,3,5,6,7] => [2,5]
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [2,4,1,5,7,3,6] => [4,2,7,3,5,1,6] => [2,3,2]
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [2,4,1,6,3,7,5] => [4,2,6,3,1,5,7] => [2,2,3]
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [2,4,1,6,7,5,3] => [4,2,1,3,7,5,6] => [2,2,3]
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [2,4,1,7,3,5,6] => [4,2,6,3,7,1,5] => [2,2,3]
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [2,4,5,3,6,7,1] => [1,2,5,3,4,6,7] => [2,5]
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [2,4,5,3,7,1,6] => [7,2,5,3,4,1,6] => [2,3,2]
[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [2,4,5,6,3,7,1] => [1,2,6,3,4,5,7] => [2,5]
[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [2,4,5,6,7,1,3] => [7,2,1,3,4,5,6] => [2,5]
[1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [2,4,5,7,3,1,6] => [7,2,6,3,4,1,5] => [2,3,2]
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [2,4,6,3,1,7,5] => [6,2,5,3,1,4,7] => [2,2,3]
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [2,4,6,3,7,5,1] => [1,2,5,3,7,4,6] => [2,2,3]
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [2,4,6,7,3,5,1] => [1,2,6,3,7,4,5] => [2,2,3]
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [2,4,7,3,1,5,6] => [6,2,5,3,7,1,4] => [2,2,3]
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [2,5,1,3,6,7,4] => [4,2,5,1,3,6,7] => [2,5]
[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [2,5,1,3,7,4,6] => [4,2,5,7,3,1,6] => [2,3,2]
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [2,5,1,6,4,7,3] => [4,2,1,6,3,5,7] => [2,1,4]
[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [2,5,1,6,7,4,3] => [4,2,1,7,3,5,6] => [2,1,4]
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [2,5,1,7,4,3,6] => [4,2,7,6,3,1,5] => [2,1,2,2]
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [2,5,6,3,4,7,1] => [1,2,5,6,3,4,7] => [2,5]
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [2,5,6,3,7,4,1] => [1,2,5,7,3,4,6] => [2,5]
[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [2,5,6,7,4,3,1] => [1,2,7,6,3,4,5] => [2,1,4]
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [2,5,7,3,4,1,6] => [7,2,5,6,3,1,4] => [2,3,2]
>>> Load all 311 entries. <<<Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
inverse
Description
Sends a permutation to its inverse.
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.
searching the database
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