A Pollak Proof for the Number of Weakly Increasing Parking Functions
Pamela E. Harris, J. Carlos Martínez Mori, Alexander N. Wilson
TL;DR
The paper tackles counting weakly increasing parking functions of length $n$, proving $|\mathrm{PF}_n^\uparrow| = C_n = \frac{1}{n+1}\binom{2n}{n}$. It introduces a Pollak-style circular-street argument that hinges on a content map $\kappa$, a weakly increasing word map $\tau$, and a rotation-based equivalence under $\omega$, with the composite $\phi = \tau \circ \kappa$ linking rotated contents to weakly increasing preferences. The core idea is that exactly one element from each content-equivalence class yields the unoccupied spot on a circular street, yielding the Catalan count $\frac{1}{n+1}\binom{2n}{n}$. This work provides a self-contained Catalan-enumeration via a cyclic-shift argument within parking-function combinatorics, complementing existing ballot and Dyck-path perspectives and connecting to broader Catalan structures.
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.