- FindStat

Identifier

Mp00320: Parking functions —reverse⟶ Parking functions
Mp00055: Parking functions —to labelling permutation⟶ Permutations
St000354: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 18247 entries. <<<

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$F_{2} = 2 + q$

$F_{3} = 5 + 10\ q + q^{2}$

$F_{4} = 14 + 73\ q + 37\ q^{2} + q^{3}$

$F_{5} = 42 + 476\ q + 651\ q^{2} + 126\ q^{3} + q^{4}$

$F_{6} = 132 + 2952\ q + 8530\ q^{2} + 4770\ q^{3} + 422\ q^{4} + q^{5}$

Description

The number of recoils of a permutation.
A recoil, or inverse descent of a permutation

$\pi$ is a value

$i$ such that

$i+1$ appears to the left of

$i$ in

$\pi_1,\pi_2,\dots,\pi_n$ .
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern

$([2,1], {(0,1),(1,1),(2,1)})$ , i.e., the middle row is shaded.

Map

to labelling permutation

Description

Returns the labelling of the support Dyck path of the parking function.

Map

reverse

Description

Return the reversal of a parking function.