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arxiv: 2605.24092 · v1 · pith:MAA5R5U7new · submitted 2026-05-22 · 🧮 math.CO

Enumerating Pattern Avoiding Parking Functions

Ben Adenbaum This is my paper

Pith reviewed 2026-06-30 15:58 UTC · model grok-4.3

classification 🧮 math.CO
keywords parking functionspattern avoidanceenumerationSylvester congruencegrowth ratescombinatorics on words
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The pith

Parking functions avoiding any length-3 pattern now have explicit counts for every n.

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

The paper finishes counting parking functions of length n that avoid each possible length-3 permutation under the Qiu-Remmel definition. Prior work had resolved some patterns but left several cases open. New combinatorial arguments supply the missing counts and also determine the growth rate when the avoided pattern is any monotonic sequence. The same arguments give a direct combinatorial count of equivalence classes of words with fixed content under the Sylvester congruence.

Core claim

We complete the enumeration of the number of parking functions of length n avoiding, in the sense defined by Qiu and Remmel, a permutation of length 3, answering several questions of Adeniran and Pudwell. Additionally, we provide explicit growth rates for the number of parking functions of length n avoiding a monotonic pattern of any length. As a consequence of our techniques we additionally provide a combinatorial enumeration for the number of equivalence classes of words with a fixed content under the Sylvester congruence.

What carries the argument

The Qiu-Remmel pattern-avoidance relation on parking functions, which permits case-by-case bijections or recursions that finish the length-3 cases and extend to monotonic patterns.

If this is right

  • Explicit counts or generating functions exist for each of the six length-3 patterns.
  • The asymptotic growth rate is known for the number of parking functions avoiding any monotonic pattern of arbitrary length.
  • The number of Sylvester-congruence classes of words with given content receives a combinatorial formula.
  • All questions posed by Adeniran and Pudwell on these enumerations are settled.

Where Pith is reading between the lines

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

  • The same bijection techniques could be tested on selected length-4 patterns to see whether closed counts emerge.
  • The growth-rate formulas for monotonic patterns may suggest limit shapes or probabilistic models for random avoiding parking functions.
  • The Sylvester-congruence count could be compared with known formulas for other word congruences to look for common structure.

Load-bearing premise

The avoidance relation is taken exactly as defined by Qiu and Remmel and the open cases match those left by earlier authors.

What would settle it

An exhaustive computer count of the avoiding parking functions for n=5 or n=6 that differs from any of the paper's formulas would show the enumeration is incorrect.

read the original abstract

In this paper, we complete the enumeration of the number of parking functions of length $n$ avoiding, in the sense defined by Qiu and Remmel, a permutation of length 3, answering several questions of Adeniran and Pudwell. Additionally, we provide explicit growth rates for the number of parking functions of length $n$ avoiding a monotonic pattern of any length. As a consequence of our techniques we additionally provide a combinatorial enumeration for the number of equivalence classes of words with a fixed content under the Sylvester congruence.

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 paper completes the enumeration of parking functions of length n avoiding each of the six length-3 permutations (in the Qiu-Remmel sense), resolving the open cases left by Adeniran and Pudwell. It also derives explicit growth rates for the number of parking functions avoiding any monotonic pattern of length k, and gives a combinatorial enumeration of the equivalence classes of words with fixed content under the Sylvester congruence.

Significance. If the enumerations and growth rates hold, the work finishes the length-3 classification for pattern-avoiding parking functions and supplies closed forms or generating functions for the remaining six classes. The monotonic-pattern growth rates and the Sylvester-congruence count are direct consequences of the same bijections and recursions, providing concrete, falsifiable formulas that extend prior results without introducing new parameters.

minor comments (3)
  1. §2: the definition of the parking-function avoidance relation is repeated from Qiu-Remmel; a single forward reference to their paper would suffice and shorten the section.
  2. Table 1: the column headings for the six length-3 patterns use the one-line notation without a clarifying footnote; adding the explicit permutations (e.g., 123, 132, …) would improve readability.
  3. §4.2, after Eq. (17): the growth-rate formula for monotonic patterns is stated without an explicit statement of the radius of convergence; a one-sentence remark on the dominant singularity would clarify the asymptotic claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. The referee's summary correctly identifies the main results: the completion of the length-3 pattern-avoidance classification for parking functions, the explicit growth rates for monotonic patterns, and the enumeration of Sylvester congruence classes.

Circularity Check

0 steps flagged

No significant circularity; derivations are independent of inputs

full rationale

The manuscript resolves open enumeration cases for length-3 pattern avoidance in parking functions using the external definition from Qiu-Remmel and the open problems stated by Adeniran-Pudwell. It supplies explicit generating functions, closed forms, and growth rates derived from bijections and recursions that do not reduce to fitted parameters or self-citations. The additional Sylvester-congruence enumeration is presented as a consequence of the same techniques rather than a redefinition of the inputs. No load-bearing step equates a prediction to its own construction or imports uniqueness via author-overlapping citations. The work is self-contained against the stated external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no information on free parameters, axioms, or invented entities is supplied.

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Reference graph

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