- FindStat

Identifier

Mp00245: Signed permutations —standardize⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions

Images

[1] => [1] => [1] => [1]

[-1] => [1] => [1] => [1]

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

[1,-2] => [1,2] => [2] => [2]

[-1,2] => [2,1] => [1,1] => [1,1]

[-1,-2] => [2,1] => [1,1] => [1,1]

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

[2,-1] => [1,2] => [2] => [2]

[-2,1] => [2,1] => [1,1] => [1,1]

[-2,-1] => [1,2] => [2] => [2]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 4282 entries. <<<

Map

standardize

Description

Return the standardization of the signed permutation, where 1 is the smallest and -1 the largest element.
Let

$\pi\in\mathfrak H_n$ be a signed permutation. Assuming the order

$1 < \dots < n < -n < \dots < -1$ , this map returns the permutation in

$\mathfrak S_n$ which is order isomorphic to

$\pi(1),\dots,\pi(n)$ .

Map

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation

$\pi$ of length

$n$ is the integer composition of

$n$ whose descent set equals the descent set of

$\pi$ . The descent set of a permutation

$\pi$ is

$\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$ . The descent set of a composition

$c = (i_1, i_2, \ldots, i_k)$ is the set

$\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$ .

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..