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Matching statistic: St001209
St001209: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[1,2] => 1
[2,1] => 0
[1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 0
[2,1,1] => 0
[1,1,3] => 1
[1,3,1] => 1
[3,1,1] => 0
[1,2,2] => 2
[2,1,2] => 1
[2,2,1] => 0
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 0
[1,1,1,1] => 0
[1,1,1,2] => 1
[1,1,2,1] => 0
[1,2,1,1] => 0
[2,1,1,1] => 0
[1,1,1,3] => 1
[1,1,3,1] => 1
[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] => 0
[1,1,2,2] => 2
[1,2,1,2] => 1
[1,2,2,1] => 0
[2,1,1,2] => 1
[2,1,2,1] => 0
[2,2,1,1] => 0
[1,1,2,3] => 2
[1,1,3,2] => 2
[1,2,1,3] => 1
[1,2,3,1] => 1
[1,3,1,2] => 2
[1,3,2,1] => 0
[2,1,1,3] => 1
[2,1,3,1] => 1
[2,3,1,1] => 0
[3,1,1,2] => 1
[3,1,2,1] => 0
Description
The pmaj statistic of a parking function.
This is the parking function analogue of the bounce statistic, see [1, Section 1.3].
The definition given there is equivalent to the following: One again and again scans the parking function from right to left, keeping track of how often one has started over again. At step $i$, one marks the next position whose value is at most $i$ with the number of restarts from the end of the parking function. The pmaj statistic is then the sum of the markings.
For example, consider the parking function $[6,2,4,1,4,1,7,3]$.
In the first round, we mark positions $6,4,2$ with $0$'s.
In the second round, we mark positions $8,5,3,1$ with $1$'s.
In the third round, we mark position $7$ with a $2$.
In total, we obtain that
$$\operatorname{pmaj}([6,2,4,1,4,1,7,3]) = 6.$$
This statistic is equidistributed with the area statistic [[St000188]] and the dinv statistic [[St000136]].
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