An Introduction to Boundaries of Groups

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= (a1,a2,…,am) 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,…,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 via a recursive formula.