Pullback parking functions

TL;DR

This work generalizes parking functions via a two-parameter pullback rule, defining $(k,\ell)$-pullback $(m,n)$-parking functions for $m\le n$ and preferences $\alpha\in[n]^m$. It develops two counting frameworks: (i) counting by parking outcome (counting through permutations), yielding $|\mathrm{PF}_{m,n}(k,\ell)|=\sum_{\pi\in\mathfrak{S}_{m,n}}\prod_{i=1}^n\big[\mathrm{B}(\pi_i)+\mathrm{F}(\pi_i)+1\big]$ with $\mathrm{B},\mathrm{F}$ encoding backtracking and forward options, and (ii) a purely recursive formula expressed in terms of smaller $|\mathrm{PF}|$ and contained counts $|\mathrm{C}_{*,*}(k,\ell)|$, via a case-based decomposition of the final car's parking choice. Special cases recover known variants: $\ell=n-1$ yields $k$-Naples $(m,n)$-parking functions, and $m=n$ recovers Christensen et al.'s results; the case $k=\ell=1$ yields vacillating parking functions. The paper also introduces contained parking functions on subintervals to support the recursion and outlines several directions for future work and generalizations. Collectively, these results advance the enumerative theory of parking-function generalizations and connect to established Naples and interval models.

Abstract

We introduce a generalization of parking functions in which cars are limited in their movement backwards and forwards by two nonnegative integer parameters $k$ and $\ell$, respectively. In this setting, there are $n$ spots on a one-way street and $m$ cars attempting to park in those spots, and $1\leq m\leq n$. We let $α=(a_1,a_2,\ldots,a_m)\in[n]^m$ denote the parking preferences for the cars, which enter the street sequentially. Car $i$ drives to their preference $a_i$ and parks there if the spot is available. Otherwise, car $i$ checks up to $k$ spots behind their preference, parking in the first available spot it encounters if any. If no spots are available, or the car reaches the start of the street, then the car returns to its preference and attempts to park in the first spot it encounters among spots $a_i+1,a_i+2,\ldots,a_i+\ell$. If car $i$ fails to park, then parking ceases. If all cars are able to park given the preferences in $α$, then $α$ is called a $(k,\ell)$-pullback $(m,n)$-parking function. Our main result establishes counts for these parking functions in two ways: counting them based on their final parking outcome (the order in which the cars park on the street), and via a recursive formula. Specializing $\ell=n-1$, our result gives a new formula for the number of $k$-Naples $(m,n)$-parking functions and further specializing $m=n$ recovers a formula for the number of $k$-Naples parking functions given by Christensen et al. The specialization of $k=\ell=1$, gives a formula for the number of vacillating $(m,n)$-parking functions, a generalization of vacillating parking functions studied by Fang et al., and the $m=n$ result answers a problem posed by the authors. We conclude with a few directions for further study.

Figures (7)

• Figure 1: Pullback toy cars. Image credit: Lucy Martinez.
• Figure 2: Parking procedure for $\alpha=(3,2,3,1)$ using the $(1,2)$-pullback parking rule.
• Figure 3: In the figure, spot $i$ is left open and the region to the left of spot $i$ consists of spots $1$ through $i-1$, and there are $x$ cars parked in those spots. Hence, $x\leq i-1$ and $x \leq m-1$. The region to the right of spot $i$ consists of the spots $i+1$ through $n$, and $m-1-x$ cars are parked in that region. Hence, $m-1-x \leq n-i$.
• Figure 4: In the figure, spot $i$ is vacant. The region to the right of spot $i$ consists of the spots numbered $i+1$ through $n$. In this case, the popular region is highlighted in red. Here, the popular region satisfies $R=n-i$. The region to the left of spot $i$ consists of the spots numbered $1$ through $i-1$, and the remaining $m-1-(n-i)$ cars are parked there.
• Figure 5: In the figure, spot $i$ is left vacant. The region to the left of spot $i$ consists of the spots numbered $1$ through $i-1$, and there are $x$ cars parked in those spots, with $x\leq i-1$. The region to the immediate right of spot $i$ consists of spots $i+1$ through $i+R$. The popular region is highlighted in red and contains exactly $R$ cars. Moreover, the spot immediately to the right of the popular region is vacant. The final region, to the far right of the street, consists of spots $i+R+2$ through $n$, and the the remaining $m-1-x-R$ cars park there such that $m-1-x-R\leq n-R-i-1$.

Theorems & Definitions (43)

• Remark 1.1
• Definition 2.1: countingthroughpermsSpiro
• Example 2.2
• Proposition 2.3
• Corollary 2.4: countingthroughperms,StanleyECVol2
• Definition 2.5
• Definition 2.6
• Example 2.7
• Definition 2.8
• Definition 2.9