TotalParkingDisplacement on ParkingFunction
The sum, over the cars, of the spots they drive past: the sum of i - p(i).
FindStat's St000188, which is also the area of the Dyck path of the parking function. Each car's displacement is its spot less its preference and the spots are 1 .. n, so the total is n (n + 1) / 2 less the preferences' sum.
Called as CombinatorialStat(x, "TotalParkingDisplacement"). The head's page has its examples.
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Distribution
Over ParkingFunctions at each size, tallied here in your browser; sizes past 6 are not offered.
FindStat St000188
The area of the Dyck path corresponding to a parking function and the total displacement of a parking function.
| Object | Value |
|---|---|
[1] | 0 |
[1,1] | 1 |
[1,2] | 0 |
[2,1] | 0 |
[1,1,1] | 3 |
[1,1,2] | 2 |
[1,1,3] | 1 |
[1,2,1] | 2 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000188 has the rest.