- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000619: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2\ q$

$F_{3} = 6\ q$

$F_{4} = 12\ q + 12\ q^{2}$

$F_{5} = 30\ q + 60\ q^{2} + 30\ q^{3}$

$F_{6} = 60\ q + 298\ q^{2} + 302\ q^{3} + 60\ q^{4}$

Description

The number of cyclic descents of a permutation.
For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is given by the number of indices

$1 \leq i \leq n$ such that

$\pi(i) > \pi(i+1)$ where we set

$\pi(n+1) = \pi(1)$ .

Map

inverse

Description

Sends a permutation to its inverse.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

Map

Foata bijection

Description

Sends a permutation to its image under the Foata bijection.
The Foata bijection

$\phi$ is a bijection on the set of words with no two equal letters. 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 (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).