- FindStat

Identifier

Mp00052: Parking functions —to non-decreasing parking function⟶ Parking functions
St001903: Parking functions ⟶ ℤ

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of fixed points of a parking function.
If $(a_1,\dots,a_n)$ is a parking function, a fixed point is an index $i$ such that $a_i = i$.
It can be shown [1] that the generating function for parking functions with respect to this statistic is
$$ \frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right). $$

Map

to non-decreasing parking function

Description

Return the non-decreasing parking function which underlies the parking function.
Sorts the parking function into an increasing sequence.