- FindStat

Identifier

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1192 entries. <<<

Map

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

Description

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

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.

Map

Foata bijection

Description

The Foata bijection for compositions.
The Foata bijection

$\phi$ is a bijection on the set of words whose letters are positive integers. It can be defined by induction on the size of the word:
Given a word

$w_1 w_2 ... w_n$ , compute the image inductively by starting with

$\phi(w_1) = w_1$ .
At the

$i$ -th step, if

$\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$ , define

$\phi(w_1 w_2 ... w_i w_{i+1})$ by placing

$w_{i+1}$ on the end of the word

$v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.

If $w_{i+1} \geq v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$ .
If $w_{i+1} < v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$ .

In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute

$\phi([1,4,2,5,3])$ , the sequence of words is

$1$
$|1|4 \to 14$
$|14|2 \to 412$
$|4|1|2|5 \to 4125$
$|4|125|3 \to 45123.$

In total, this gives

$\phi([1,4,2,5,3]) = [4,5,1,2,3]$ .
This bijection sends the major index St000769The major index of a composition regarded as a word. to the number of inversions St000766The number of inversions of an integer composition..