Parking Functions: From Combinatorics to Probability

Suppose that m drivers each choose a preferred parking space in linear car park with n spots. In order, driver goes to their chosen spot and parks there if possible, otherwise takes the next available it exists. If all successfully, sequence of choices is called function. Classical functions correspond case $$m=n$$ ; we study here combinatorial probabilistic aspects this generalized case. We construct family bijections between $$\text {PF}(m, n)$$ cars spots spanning forests $$\mathscr {F}(n+1, n+1-m)$$ $$n+1$$ vertices $$n+1-m$$ distinct trees having specified roots. This leads bijective correspondence monomial terms associated Tutte polynomial disjoint union $$n-m+1$$ complete graphs. present an identity “inversion enumerator” fixed roots “displacement functions. The displacement then related number graphs on labeled edges, where graph has rooted components investigate various properties uniform function, giving formula for law single coordinate. As side result obtain recurrence relation enumerator. Adapting known results random probes, further deduce covariance two coordinates when .