- FindStat

Identifier

Mp00053: Parking functions —to car permutation⟶ Permutations
St001298: Permutations ⟶ ℤ

Values

[1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 18248 entries. <<<

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$F_{1} = 1$

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

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

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

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

$F_{6} = 1 + 226\ q + 2982\ q^{2} + 7741\ q^{3} + 5137\ q^{4} + 720\ q^{5}$

Description

The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation

$\pi$ is the sequence

$(v_1,\dots,v_n)$ , with

$v_i=|\{j > i: \pi(j) < \pi(i)\}$ . This statistic counts the number of distinct elements in this sequence.

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.