  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.
