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Identifier

St001903: Parking functions ⟶ ℤ

Values

[1] => 1

[1,1] => 1

[1,2] => 2

[2,1] => 0

[1,1,1] => 1

[1,1,2] => 1

[1,2,1] => 2

[2,1,1] => 0

[1,1,3] => 2

[1,3,1] => 1

[3,1,1] => 0

[1,2,2] => 2

[2,1,2] => 0

[2,2,1] => 1

[1,2,3] => 3

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 0

[3,1,2] => 0

[3,2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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
$$ \frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right). $$

References

[1] Richard Stanley and Mei Yin, 2023, private communication.

Code

def statistic(p):
    return sum(1 for i, e in enumerate(p, 1) if i == e)

def generating_function(n):
    R. = ZZ[]
    return 1/(n+1)^2*((q+n)^(n+1) - (q-1)^(n+1))

Created

Jun 08, 2023 at 09:00 by Martin Rubey

Updated

Jun 15, 2023 at 14:41 by Martin Rubey