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arxiv: 2604.06735 · v1 · submitted 2026-04-08 · 🧮 math.CO

Simultaneous avoidance of length-4 patterns in ascent sequences

Qi Liu , Sergey Kitaev , Philip B. Zhang This is my paper

Pith reviewed 2026-05-10 18:07 UTC · model grok-4.3

classification 🧮 math.CO
keywords ascent sequencespattern avoidanceWilf equivalenceFishburn numbersCatalan numbersFibonacci numbersgenerating treesrestricted growth functions
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The pith

Ascent sequences avoiding subsets of five length-4 patterns fall into 16 Wilf equivalence classes with explicit enumerations.

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

The paper studies ascent sequences, objects in bijection with Fishburn families such as (2+2)-free posets. It considers simultaneous avoidance of the five length-4 patterns 0101, 0102, 0112, 0120 and 0121. For all but three of the 32 possible subsets, the authors supply complete enumerations obtained through decompositions and generating trees. The resulting classes exhibit a range of behaviors, including matches to the Catalan and Fibonacci sequences, polynomial counts, and rational generating functions. Three avoidance sets are left open.

Core claim

Ascent sequences avoiding any subset of the patterns 0101, 0102, 0112, 0120, and 0121 fall into 16 Wilf equivalence classes. With the exception of the open cases of avoiding 0120, avoiding 0121, and avoiding both, the authors give explicit enumerations or generating functions for each class by combining structural decompositions, generating-tree constructions, and reductions to shorter patterns via restricted growth functions.

What carries the argument

Structural decompositions of ascent sequences combined with generating-tree techniques that track the avoidance constraints, together with reductions to shorter patterns using restricted growth functions.

If this is right

  • Sixteen distinct Wilf classes arise from the 32 possible avoidance sets.
  • Several classes are counted by the Catalan numbers or the Fibonacci numbers.
  • Other classes have polynomial enumerations or rational generating functions.
  • The methods reduce certain length-4 avoidances to known shorter-pattern cases.
  • The results extend pattern-avoidance studies from ascent sequences to related Fishburn objects.

Where Pith is reading between the lines

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

  • The three open cases may admit similar decompositions once their minimal obstructions are better understood.
  • The Wilf classes found here likely correspond to avoidance classes in Stoimenow matchings and (2+2)-free posets.
  • The new sequences encountered could be checked for further combinatorial interpretations such as lattice paths or plane trees.
  • The reduction technique via restricted growth functions may generalize to longer patterns or other pattern sets.

Load-bearing premise

The decompositions and generating trees used for each avoidance set correctly and exhaustively list every valid ascent sequence without missing cases or double-counting.

What would settle it

An ascent sequence that avoids one of the studied pattern subsets yet whose position in the enumeration does not match the claimed count or generating function for that Wilf class.

read the original abstract

Ascent sequences form a central class of combinatorial objects, as they are in bijection with several important families such as (2+2)-free posets, Stoimenow matchings, and other Fishburn objects, and are enumerated by the Fishburn numbers. We study pattern avoidance in ascent sequences for the five patterns of length 4: $0101$, $0102$, $0112$, $0120$, and $0121$. These patterns arise naturally from recent work on pattern avoidance in related families of Fishburn objects, including Stoimenow matchings and (2+2)-free posets. We enumerate ascent sequences avoiding any subset of these patterns, with the exception of the sets $\{0120\}$, $\{0121\}$, and $\{0120,0121\}$, for which the enumeration remains open. Our results reveal that the corresponding avoidance classes fall into $16$ Wilf equivalence classes and exhibit a wide range of enumerative behaviour, including connections to classical sequences such as the Catalan and Fibonacci numbers, as well as polynomial formulas and rational generating functions; several of the sequences we obtain appear to be new. Our methods combine structural decompositions with generating-tree techniques and, in several cases, rely on reductions to shorter patterns via restricted growth functions. This work contributes to the broader study of pattern avoidance across Fishburn families and highlights further connections between ascent sequences and other combinatorial structures.

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 / 0 minor

Summary. The paper enumerates ascent sequences avoiding any subset of the five length-4 patterns 0101, 0102, 0112, 0120, and 0121, except for the three open cases {0120}, {0121}, and {0120,0121}. It shows that the avoidance classes fall into 16 Wilf equivalence classes with enumerations ranging from Catalan and Fibonacci numbers to polynomials and rational generating functions. Methods combine structural decompositions, generating trees, and restricted-growth-function reductions to shorter patterns.

Significance. If the results hold, the work provides a near-complete classification of simultaneous avoidance for these patterns in ascent sequences, a central Fishburn family. It reveals diverse enumerative behaviors and new sequences while properly flagging the three open cases, strengthening connections between ascent-sequence avoidance and related structures such as (2+2)-free posets and Stoimenow matchings. The use of standard, previously validated techniques in the literature adds reliability to the classification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on the 16 Wilf classes and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity: enumerations from explicit decompositions and trees

full rationale

The paper derives its enumerations of ascent-sequence avoidance classes via structural decompositions, generating-tree constructions, and reductions to shorter patterns using restricted growth functions. These techniques are applied directly to the five length-4 patterns and their subsets, producing explicit counts that match known sequences (Catalan, Fibonacci) or yield polynomials/rational GFs without any fitted parameters, self-definitional loops, or load-bearing self-citations. The three open cases are stated as such rather than forced closed. The derivation chain remains independent of its own outputs and is self-contained against standard combinatorial benchmarks in the Fishburn/pattern-avoidance literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on established combinatorial structures and methods without introducing new free parameters or entities.

axioms (2)
  • domain assumption Ascent sequences are in bijection with (2+2)-free posets, Stoimenow matchings, and other Fishburn objects.
    Invoked to motivate the patterns from recent work on related families.
  • standard math Structural decompositions and generating-tree techniques apply to pattern avoidance in ascent sequences.
    Core enumeration method used throughout.

pith-pipeline@v0.9.0 · 5549 in / 1490 out tokens · 70064 ms · 2026-05-10T18:07:04.479144+00:00 · methodology

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Reference graph

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