4 research outputs found
Permutation Invariant Parking Assortments
We introduce parking assortments, a generalization of parking functions with
cars of assorted lengths. In this setting, there are cars of
lengths entering a one-way
street with parking spots. The cars have parking
preferences , where
, and enter the street in order. Each car ,
with length and preference , follows a natural extension of the
classical parking rule: it begins looking for parking at its preferred spot
and parks in the first contiguously available spots thereafter, if
there are any. If all cars are able to park under the preference list
, we say is a parking assortment for .
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 for
, we say it is permutation invariant if all rearrangements of
are also parking assortments for . While all parking
functions are permutation invariant, this is not the case for parking
assortments in general, motivating the need for a characterization of this
property. Although obtaining a full characterization for arbitrary
and remains elusive, we do so for
. 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 .
Our results around minimally invariant car lengths also hold for parking
sequences
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum