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arxiv: 2511.20796 · v3 · submitted 2025-11-25 · 🧮 math.CO

A Pollak Proof for the Number of Weakly Increasing Parking Functions

Pamela E. Harris , J. Carlos Mart\'inez Mori , Alexander N. Wilson This is my paper

Pith reviewed 2026-05-17 04:06 UTC · model grok-4.3

classification 🧮 math.CO
keywords weakly increasing parking functionsCatalan numbersPollak argumentcircular street constructioncombinatorial enumerationnon-decreasing sequencesbijection
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The pith

A circular-street argument proves the number of weakly increasing parking functions of length n equals the nth Catalan number.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a new combinatorial proof that weakly increasing parking functions of length n are counted by the Catalan number C_n = 1/(n+1) times binom(2n, n). It adapts a circular-street construction in the style of Pollak to count these non-decreasing sequences that satisfy the parking function condition. A reader would care because this directly links a restricted class of parking functions to the many other objects enumerated by Catalan numbers, offering a visual or geometric way to see the equality without relying on generating functions or recursion.

Core claim

We develop a circular-street argument, in the style of Pollak, to obtain a new proof that there are C_n = 1/(n+1) binom(2n,n) weakly increasing parking functions of length n >= 1, where C_n is the nth Catalan number.

What carries the argument

The circular-street argument, which arranges cars and spots in a circle to establish an exact count or bijection for non-decreasing sequences that meet the parking condition.

If this is right

  • The total count of weakly increasing parking functions of length n is exactly the nth Catalan number for every n at least 1.
  • This supplies a direct counting proof that avoids generating functions or inductive arguments used in prior work.
  • The same circular construction can be applied to verify the enumeration for any fixed n by explicit mapping of sequences to valid parking configurations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The argument may adapt to count other restricted parking functions, such as those with additional monotonicity conditions.
  • It suggests a geometric interpretation that could connect parking functions to lattice paths or Dyck words counted by the same Catalan formula.
  • Researchers might use the circular model to derive similar enumerations for labeled or colored variants of these sequences.

Load-bearing premise

The circular-street construction correctly establishes a bijection or exact count for the weakly increasing condition without missing cases or introducing overcounts when the sequence is restricted to be non-decreasing.

What would settle it

For n=3, where the Catalan number is 5, enumerate all weakly increasing sequences of length 3 and check whether the circular-street mapping produces exactly five distinct objects without omissions or duplicates.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a circular-street argument in the style of Pollak to prove that the number of weakly increasing parking functions of length n is the nth Catalan number C_n = 1/(n+1) binom(2n,n).

Significance. If the central argument holds, the paper supplies a direct combinatorial counting proof for a known enumeration result on weakly increasing parking functions. It extends Pollak's rotation-based method from the unrestricted case to the non-decreasing restriction and avoids parameter fitting or recursive definitions, which is a strength for combinatorial clarity.

minor comments (3)
  1. [Introduction] The abstract and introduction would benefit from a short explicit statement of how the circular construction selects non-decreasing representatives (e.g., via the choice of the 'last car' position) to make the adaptation from Pollak's original argument immediately visible.
  2. [§3] A small illustrative example for n=3 showing the circular arrangements and the resulting weakly increasing sequences would improve readability of the main construction.
  3. [References] The paper should include a reference to Pollak's original 1970s work on parking functions to clarify the precise technical differences in the restricted case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. The report correctly identifies the paper's contribution as a direct Pollak-style circular-street argument establishing that the number of weakly increasing parking functions of length n equals the nth Catalan number, extending the classical unrestricted case while preserving combinatorial clarity without recursion or auxiliary parameters.

Circularity Check

0 steps flagged

Direct combinatorial construction with no reduction to inputs

full rationale

The paper presents a new proof via a circular-street argument adapted from Pollak to directly establish that the number of weakly increasing parking functions of length n equals the nth Catalan number. This is a self-contained combinatorial counting or bijection argument rather than any derivation that reduces the target count to itself by construction, fitted parameters renamed as predictions, or load-bearing self-citations. No equations or steps in the described chain equate the result to its own inputs; the construction is offered as an independent verification of the known enumeration. Per the guidelines, a direct combinatorial proof without the enumerated circularity patterns receives score 0 and an empty steps list.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on the standard definition of parking functions and the combinatorial interpretation of Catalan numbers; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math The classical definition of (weakly increasing) parking functions and the known formula for the nth Catalan number.
    Invoked when stating the target count C_n.

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