In the summer of 2024, I participated in a Research Experience for Undergraduates (REU) funded by the NSF in Ponce, Puerto Rico. I worked on and co-authored two papers throughout the experience, one of which is now published in the American Journal of Combinatorics. The other is now available in preprint!

Pullback Parking Functions

https://doi.org/10.63151/amjc.v4i.21

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 ℓ, respectively. In this setting, there are n spots on a one-way street and m cars attempting to park in those spots, and 1 ≤ m ≤ n. We let α = (a_1, a_2, . . . , am) ∈ [n]^m denote the parking preferences for the cars, which enter the street sequentially. Car i drives to their preference ai 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, . . . , a_i + ℓ. 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, ℓ)-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 ℓ = 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. [5]. The specialization of k = ℓ = 1, gives a formula for the number of vacillating (m, n)-parking functions, a generalization of vacillating parking functions studied by Fang et al. [8], and the m = n result answers a problem posed by the authors. We conclude with a few directions for further study.

Lehmer Parking Functions and Their Outcomes

https://arxiv.org/html/2603.20535v1

Abstract:

We introduce Lehmer parking functions and study their set of parking outcomes. Our main results establish that the number of outcomes of Lehmer parking functions of length n is given by a Bell number, which is exactly the number of set partitions of an n element set. We also show that the number of outcomes of weakly decreasing Lehmer parking functions is given by a Catalan number, which corresponds to a subset of set partitions on a set with n elements referred to as non-intersecting set partitions.