A Pollak Proof for the Number of Weakly Increasing Parking Functions

TL;DR

This paper presents a new Pollak-style circular-street proof demonstrating that the number of weakly increasing parking functions of length n equals the nth Catalan number.

Contribution

It introduces a novel circular-street argument to prove the enumeration of weakly increasing parking functions, providing a new combinatorial proof.

Findings

Confirmed the count of weakly increasing parking functions matches Catalan numbers

Developed a Pollak-style proof technique for parking function enumeration

Provided a new combinatorial perspective on parking functions

Abstract

We develop a circular-street argument, in the style of Pollak, to obtain a new proof that there are $C_n = \frac{1}{n+1}\binom{2n}{n}$ weakly increasing parking functions of length $n \geq 1$, where $C_n$ is the $n$th Catalan number.