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Matching statistic: St001903
Mapping
St001903: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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 $$ \frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right). $$