A Short Proof that the number of $(a,b)$-parking functions of length n $a(a+bn)^{n-1}$

TL;DR

This paper presents a concise proof that the count of (a,b)-parking functions of length n is given by a(a+bn)^{n-1}, simplifying previous lengthy proofs and referencing prior foundational work.

Contribution

It provides a significantly shorter proof of a known combinatorial formula for (a,b)-parking functions, improving clarity and accessibility.

Findings

Short proof of the formula for (a,b)-parking functions

Simplifies previous complex proofs from 2003

Connects to prior work by Stanley and others

Abstract

We give a very short proof of the fact that the number of $(a,b)$-parking functions of length $n$ equals $a(a+bn)^{n-1}$. This was first proved in 2003 by Kung and Yan, via a very long and torturous route, as a corollary of a more general result. This new version contains a reference to previous work kindly communicated by Richard Stanley