Counts of subfamilies for a given graph

Determine the counts of specified subfamilies of largest neighbor parking functions for a given labeled graph, including subfamilies defined by weakly increasing preferences or by the number of lucky cars or spots.

Background

For graphs containing the path graph, the paper shows that weakly decreasing largest neighbor parking functions coincide with weakly decreasing classical parking functions. It then asks for analogous enumeration results for other classes of largest neighbor parking functions, specifically mentioning weakly increasing sequences and sequences classified by the number of lucky cars or spots.

References

Given a graph $G$, what do the counts of subfamilies of largest neighbor parking functions look like?

— Parking with Frustrated Drivers  (2609.35638 - Hallam et al., 28 Sep 2026) in Section 'Open Questions', Question following Proposition \ref{prop:neighborWeaklyDec}