- FindStat

Identifier

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions

Images

[1,0] => [1] => [1] => [1]

[1,0,1,0] => [2,1] => [1,2] => [2]

[1,1,0,0] => [1,2] => [2,1] => [2]

[1,0,1,0,1,0] => [2,3,1] => [1,2,3] => [3]

[1,0,1,1,0,0] => [2,1,3] => [3,2,1] => [2,1]

[1,1,0,0,1,0] => [1,3,2] => [2,1,3] => [3]

[1,1,0,1,0,0] => [3,1,2] => [3,1,2] => [3]

[1,1,1,0,0,0] => [1,2,3] => [2,3,1] => [3]

[1,0,1,0,1,0,1,0] => [2,3,4,1] => [1,2,3,4] => [4]

[1,0,1,0,1,1,0,0] => [2,3,1,4] => [4,2,3,1] => [3,1]

[1,0,1,1,0,0,1,0] => [2,1,4,3] => [3,2,1,4] => [2,2]

[1,0,1,1,0,1,0,0] => [2,4,1,3] => [4,2,1,3] => [2,2]

[1,0,1,1,1,0,0,0] => [2,1,3,4] => [3,2,4,1] => [2,2]

[1,1,0,0,1,0,1,0] => [1,3,4,2] => [2,1,3,4] => [4]

[1,1,0,0,1,1,0,0] => [1,3,2,4] => [2,4,3,1] => [3,1]

[1,1,0,1,0,0,1,0] => [3,1,4,2] => [3,1,2,4] => [4]

[1,1,0,1,0,1,0,0] => [3,4,1,2] => [4,1,2,3] => [4]

[1,1,0,1,1,0,0,0] => [3,1,2,4] => [3,4,2,1] => [3,1]

[1,1,1,0,0,0,1,0] => [1,2,4,3] => [2,3,1,4] => [4]

[1,1,1,0,0,1,0,0] => [1,4,2,3] => [2,4,1,3] => [4]

[1,1,1,0,1,0,0,0] => [4,1,2,3] => [3,4,1,2] => [4]

[1,1,1,1,0,0,0,0] => [1,2,3,4] => [2,3,4,1] => [4]

[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [1,2,3,4,5] => [5]

[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [5,2,3,4,1] => [4,1]

[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [4,2,3,1,5] => [3,2]

[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [5,2,3,1,4] => [3,2]

[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [4,2,3,5,1] => [3,2]

[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [3,2,1,4,5] => [2,3]

[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [3,2,5,4,1] => [2,2,1]

[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [4,2,1,3,5] => [2,3]

[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [5,2,1,3,4] => [2,3]

[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [4,2,5,3,1] => [2,2,1]

[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [3,2,4,1,5] => [2,3]

[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [3,2,5,1,4] => [2,3]

[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [4,2,5,1,3] => [2,3]

[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [3,2,4,5,1] => [2,3]

[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [2,1,3,4,5] => [5]

[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [2,5,3,4,1] => [4,1]

[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [2,4,3,1,5] => [3,2]

[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [2,5,3,1,4] => [3,2]

[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [2,4,3,5,1] => [3,2]

[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [3,1,2,4,5] => [5]

[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [3,5,2,4,1] => [4,1]

[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [4,1,2,3,5] => [5]

[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => [5]

[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [4,5,2,3,1] => [4,1]

[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [3,4,2,1,5] => [3,2]

[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [3,5,2,1,4] => [3,2]

[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [4,5,2,1,3] => [3,2]

[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [3,4,2,5,1] => [3,2]

[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [2,3,1,4,5] => [5]

[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [2,3,5,4,1] => [4,1]

[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [2,4,1,3,5] => [5]

[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [2,5,1,3,4] => [5]

[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [2,4,5,3,1] => [4,1]

[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [3,4,1,2,5] => [5]

[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [3,5,1,2,4] => [5]

[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [4,5,1,2,3] => [5]

[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [3,4,5,2,1] => [4,1]

[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [2,3,4,1,5] => [5]

[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [2,3,5,1,4] => [5]

[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [2,4,5,1,3] => [5]

[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [3,4,5,1,2] => [5]

[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [2,3,4,5,1] => [5]

[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [6]

[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [6,2,3,4,5,1] => [5,1]

[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [5,2,3,4,1,6] => [4,2]

[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => [4,2]

[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [5,2,3,4,6,1] => [4,2]

[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [4,2,3,1,5,6] => [3,3]

[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [4,2,3,6,5,1] => [3,2,1]

[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [5,2,3,1,4,6] => [3,3]

[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [6,2,3,1,4,5] => [3,3]

[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [5,2,3,6,4,1] => [3,2,1]

[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [4,2,3,5,1,6] => [3,3]

[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [4,2,3,6,1,5] => [3,3]

[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [5,2,3,6,1,4] => [3,3]

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [4,2,3,5,6,1] => [3,3]

[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [3,2,1,4,5,6] => [2,4]

[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [3,2,6,4,5,1] => [2,3,1]

[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [3,2,5,4,1,6] => [2,2,2]

[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => [3,2,6,4,1,5] => [2,2,2]

[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [3,2,5,4,6,1] => [2,2,2]

[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [4,2,1,3,5,6] => [2,4]

[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [4,2,6,3,5,1] => [2,3,1]

[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [5,2,1,3,4,6] => [2,4]

[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [6,2,1,3,4,5] => [2,4]

[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [5,2,6,3,4,1] => [2,3,1]

[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [4,2,5,3,1,6] => [2,2,2]

[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [4,2,6,3,1,5] => [2,2,2]

[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [5,2,6,3,1,4] => [2,2,2]

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [4,2,5,3,6,1] => [2,2,2]

[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [3,2,4,1,5,6] => [2,4]

[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [3,2,4,6,5,1] => [2,3,1]

[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [3,2,5,1,4,6] => [2,4]

[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [3,2,6,1,4,5] => [2,4]

[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [3,2,5,6,4,1] => [2,3,1]

[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [4,2,5,1,3,6] => [2,4]

[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [4,2,6,1,3,5] => [2,4]

[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [5,2,6,1,3,4] => [2,4]

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [4,2,5,6,3,1] => [2,3,1]

>>> Load all 951 entries. <<<

Map

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.

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.