- FindStat

Identifier

Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000662: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 18248 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 1 + 2\ q$

$F_{3} = 1 + 9\ q + 6\ q^{2}$

$F_{4} = 1 + 29\ q + 71\ q^{2} + 24\ q^{3}$

$F_{5} = 1 + 83\ q + 511\ q^{2} + 581\ q^{3} + 120\ q^{4}$

$F_{6} = 1 + 226\ q + 2998\ q^{2} + 7717\ q^{3} + 5145\ q^{4} + 720\ q^{5}$

Description

The staircase size of the code of a permutation.
The code

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

$(c_1,\ldots,c_{n})$ with

$c_i = |\{j > i : \pi(j) < \pi(i)\}|$ . This is a bijection between permutations and all sequences

$(c_1,\ldots,c_n)$ with

$0 \leq c_i \leq n-i$ .
The staircase size of the code is the maximal

$k$ such that there exists a subsequence

$(c_{i_k},\ldots,c_{i_1})$ of

$c(\pi)$ with

$c_{i_j} \geq j$ .
This statistic is mapped through Mp00062Lehmer-code to major-code bijection to the number of descents, showing that together with the number of inversions St000018The number of inversions of a permutation. it is Euler-Mahonian.

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

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

$\tau(i) = n+1-\sigma(i)$