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Matching statistic: St000942
Mapping
St000942: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 0
[1,2] => 2
[2,1] => 1
[1,1,1] => 0
[1,1,2] => 0
[1,2,1] => 0
[2,1,1] => 0
[1,1,3] => 1
[1,3,1] => 1
[3,1,1] => 1
[1,2,2] => 1
[2,1,2] => 0
[2,2,1] => 0
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 1
[1,1,1,1] => 0
[1,1,1,2] => 0
[1,1,2,1] => 0
[1,2,1,1] => 0
[2,1,1,1] => 0
[1,1,1,3] => 0
[1,1,3,1] => 0
[1,3,1,1] => 0
[3,1,1,1] => 0
[1,1,1,4] => 1
[1,1,4,1] => 1
[1,4,1,1] => 1
[4,1,1,1] => 1
[1,1,2,2] => 0
[1,2,1,2] => 0
[1,2,2,1] => 0
[2,1,1,2] => 0
[2,1,2,1] => 0
[2,2,1,1] => 0
[1,1,2,3] => 0
[1,1,3,2] => 0
[1,2,1,3] => 0
[1,2,3,1] => 0
[1,3,1,2] => 0
[1,3,2,1] => 0
[2,1,1,3] => 0
[2,1,3,1] => 0
[2,3,1,1] => 0
[3,1,1,2] => 0
[3,1,2,1] => 0
Description
The number of critical left to right maxima of the parking functions. An entry $p$ in a parking function is critical, if there are exactly $p-1$ entries smaller than $p$ and $n-p$ entries larger than $p$. It is a left to right maximum, if there are no larger entries before it. This statistic allows the computation of the Tutte polynomial of the complete graph $K_{n+1}$, via $$ \sum_{P} x^{st(P)}y^{\binom{n+1}{2}-\sum P}, $$ where the sum is over all parking functions of length $n$, see [1, thm.13.5.16].