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Identifier

St001209: Parking functions ⟶ ℤ

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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 St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. and the dinv statistic St000136The dinv of a parking function..

References

[1] Loehr, N. A. Combinatorics of $q$, $t$-parking functions MathSciNet:2110560

Code

def statistic(PF):
    PF = list(PF)
    PF.reverse()
    n = len(PF)
    count = 1
    run   = 0
    pm    = 0
    while any( not x is Infinity for x in PF ):
        for i in range(n):
            if PF[i] <= count:
                count += 1
                PF[i] = Infinity
                pm += run
        run += 1
    return pm

Created

Jun 14, 2018 at 11:31 by Christian Stump

Updated

Jun 14, 2018 at 11:31 by Christian Stump