Enumerating Parking Completions Using Join and Split

TL;DR

This paper introduces a novel combinatorial framework using Join and Split operations to enumerate parking completions, linking them to lattice paths, polytopes, and parking functions, thus advancing understanding in combinatorics and geometric enumeration.

Contribution

It presents a new method for enumerating parking completions via Join and Split operations, connecting these to lattice paths, polytopes, and parking functions, and deriving new volume formulas.

Findings

Derived formulas for ordered and unordered parking completions.

Connected parking completions to lattice path enumeration.

Provided new volume formulas for Pitman-Stanley polytopes.

Abstract

Given a strictly increasing sequence $\mathbf{t}$ with entries from $[n]:=\{1,\ldots,n\}$, a parking completion is a sequence $\mathbf{c}$ with $|\mathbf{t}|+|\mathbf{c}|=n$ and $|\{t\in \mathbf{t}\mid t\le i\}|+|\{c\in \mathbf{c}\mid c\le i\}|\ge i$ for all $i$ in $[n]$. We can think of $\mathbf{t}$ as a list of spots already taken in a street with $n$ parking spots and $\mathbf{c}$ as a list of parking preferences where the $i$-th car attempts to park in the $c_i$-th spot and if not available then proceeds up the street to find the next available spot, if any. A parking completion corresponds to a set of preferences $\mathbf{c}$ where all cars park. We relate parking completions to enumerating restricted lattice paths and give formulas for both the ordered and unordered variations of the problem by use of a pair of operations termed \textbf{Join} and \textbf{Split}. Our results give a new volume formula for most Pitman-Stanley polytopes, and enumerate the signature parking functions of Ceballos and Gonz\'alez D'Le\'on.