- FindStat

Identifier

Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000355: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 3$

$F_{3} = 14 + 2\ q$

$F_{4} = 88 + 26\ q + 11\ q^{2}$

$F_{5} = 670 + 321\ q + 207\ q^{2} + 80\ q^{3} + 18\ q^{4}$

$F_{6} = 5762 + 4140\ q + 3356\ q^{2} + 1920\ q^{3} + 1167\ q^{4} + 297\ q^{5} + 153\ q^{6} + 12\ q^{7}$

Description

The number of occurrences of the pattern 21-3.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern

$21\!\!-\!\!3$ .

Map

to car permutation

Description

Maps a parking function to the permutation of cars that correspond to that parking function.
For example, the image

$[2, 4, 5, 6, 3, 1, 7]$ means that car 2 takes spots 1, car 4 takes spot 2, and so on.

Map

ones to leading

Description

The unique permutation obtained by applying the Foata-Riordan map to obtain a Prüfer code, then prepending zero and cyclically shifting.
Let

$c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let

$c_0 = 0$ .
This map returns the a unique permutation

$q_1,\dots, q_n$ such that

$q_i - c_{i-1}$ is constant modulo

$n+1$ .
This map is Mp00299ones to leading restricted to permutations.

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.).