Permutation Invariant Parking Assortments

We introduce parking assortments, a generalization of parking functions with cars of assorted lengths. In this setting, there are $n\in\mathbb{N}$ cars of lengths $\mathbf{y}=(y_1,y_2,\ldots,y_n)\in\mathbb{N}^n$ entering a one-way street with $m=\sum_{i=1}^ny_i$ parking spots. The cars have parking preferences $\mathbf{x}=(x_1,x_2,\ldots,x_n)\in[m]^n$, where $[m]:=\{1,2,\ldots,m\}$, and enter the street in order. Each car $i \in [n]$, with length $y_i$ and preference $x_i$, follows a natural extension of the classical parking rule: it begins looking for parking at its preferred spot $x_i$ and parks in the first $y_i$ contiguously available spots thereafter, if there are any. If all cars are able to park under the preference list $\mathbf{x}$, we say $\mathbf{x}$ is a parking assortment for $\mathbf{y}$. Parking assortments also generalize parking sequences, introduced by Ehrenborg and Happ, since each car seeks for the first contiguously available spots it fits in past its preference. Given a parking assortment $\mathbf{x}$ for $\mathbf{y}$, we say it is permutation invariant if all rearrangements of $\mathbf{x}$ are also parking assortments for $\mathbf{y}$. While all parking functions are permutation invariant, this is not the case for parking assortments in general, motivating the need for characterization of this property. Although obtaining a full characterization for arbitrary $n\in\mathbb{N}$ and $\mathbf{y}\in\mathbb{N}^n$ remains elusive, we do so for $n=2,3$. Given the technicality of these results, we introduce the notion of minimally invariant car lengths, for which the only invariant parking assortment is the all ones preference list. We provide a concise, oracle-based characterization of minimally invariant car lengths for any $n\in\mathbb{N}$. Our results around minimally invariant car lengths also hold for parking sequences.