Identifier
Values
[1] => [1] => [1] => 1
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 2
[1,2,3] => [2,3,1] => [2,3,1] => 0
[1,3,2] => [3,2,1] => [3,2,1] => 1
[2,1,3] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 3
[3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 1
[1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 2
[1,4,2,3] => [3,4,2,1] => [3,4,2,1] => 0
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 0
[2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 1
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4
[2,4,1,3] => [1,4,2,3] => [1,4,2,3] => 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,2,4] => [3,1,4,2] => [3,1,4,2] => 0
[3,1,4,2] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 0
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 2
[3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
[3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 1
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[4,1,3,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [2,3,1,4] => [2,3,1,4] => 1
[4,3,1,2] => [4,2,1,3] => [4,2,1,3] => 1
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 2
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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
Inverse Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
Map
parking function
Description
Interpret the permutation as a parking function.