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arxiv: 2605.19429 · v1 · pith:DCATF5LJnew · submitted 2026-05-19 · 🧮 math.CO

Equidistribution of mesh patterns of short length

Xinyu Su , Sergey Kitaev , Jiahao Zhang This is my paper

Pith reviewed 2026-05-20 04:34 UTC · model grok-4.3

classification 🧮 math.CO
keywords mesh patternsequidistributionWilf classespermutation patternspattern avoidanceenumerative combinatoricsgenerating functionsbijections
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The pith

Mesh patterns of length 2 fall into between 105 and 108 equidistribution classes, conjectured to be exactly 105.

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

The paper classifies how mesh patterns of length 2 distribute across permutations by showing they belong to a number of equidistribution classes that sits between 105 and 108. It conjectures the precise count is 105 after using bijections, generating functions, recurrences, and symmetries to connect and equate many of the patterns. This work also ties numerous patterns to earlier known distributions and solves seven open counting problems for avoided patterns. If the conjecture is correct, the number of distinct Wilf classes drops to an upper bound of 49, down from the earlier 56, leaving only three undecided merges. Readers interested in permutation combinatorics would care because the result organizes a large set of objects into fewer families and makes future avoidance counts more tractable.

Core claim

We show that the number of equidistribution equivalence classes lies between 105 and 108, and conjecture that it is exactly 105. As a consequence, we obtain an upper bound of 49 Wilf-classes, improving the previously known bound of 56, and reducing the problem to three remaining conjectural equivalences with the actual number conjectured to be 46. The proofs rely on bijective constructions, generating functions, recurrence relations, and structural symmetries that establish four new distribution classes and resolve seven open avoidance enumeration questions.

What carries the argument

Equidistribution equivalence classes on the set of mesh patterns of length 2, constructed and verified through explicit bijections between counting sequences together with matching generating functions and recurrence relations.

If this is right

  • Four previously unknown distribution classes receive explicit equidistribution proofs.
  • Many mesh patterns are shown to share the same distribution as patterns already studied in the literature.
  • Seven open problems on the number of permutations avoiding a given mesh pattern are settled.
  • The upper bound on the number of distinct Wilf classes is lowered from 56 to 49.
  • The remaining task reduces to deciding three specific conjectural equivalences.

Where Pith is reading between the lines

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

  • The same combination of bijections and symmetries might classify mesh patterns of length 3 once the length-2 case is settled.
  • The resolved avoidance counts suggest that many mesh patterns share avoidance numbers without requiring separate computations.
  • Automated search for additional bijections could confirm or refute the three open merges in reasonable time.
  • The classification supplies a template for grouping patterns by their statistics rather than by avoidance alone.

Load-bearing premise

The bijective constructions, generating functions, and structural symmetries are assumed to have found every true equidistribution and to have introduced no missed links or incorrect groupings among the 108 starting candidates.

What would settle it

An explicit pair of patterns placed in different classes whose permutation sets are shown to have identical counts, or a counterexample to one of the three remaining conjectures, would fix the exact number of classes.

Figures

Figures reproduced from arXiv: 2605.19429 by Jiahao Zhang, Sergey Kitaev, Xinyu Su.

Figure 1
Figure 1. Figure 1: Equidistributions in Lemma 3.1. establishes the joint equidistribution of the patterns in question. Lemma 3.1. There are two classes of equidistributed mesh patterns, shown in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Related to the proof of Theorem 3.4. Proof. The equivalence of the patterns in Class 25 follows from Lemma 3.1 applied to the patterns and . Let p = . Referring to [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Related to the proof of Theorem 3.16. Proof. The equivalence of the patterns in Class 78 follows from Lemma 3.1 applied to the patterns , , and . The distribution of is given by [6, Cor 2.5]. Only part of the proof of the next theorem has an “easily explainable equidistribution”. Theorem 3.16. The patterns in Class 80 are equivalent. Moreover, A(x) = (1 + x)F(x) 1 + xF(x) , F(x, q) = (1 + x(1 − q))F(x) 1 +… view at source ↗
Figure 4
Figure 4. Figure 4: Related to the proof of Theorem 4.1. Theorem 5.1. The patterns in Class 15 are equivalent. For any pattern p in the class, sn,k(p) = sn−1,k(p) + (n − 1)sn−1,k−1(p) (8) with the initial conditions s0,0(p) = s1,0 = 1 and s1,1(p) = 0, which shows that sn,k(p) = C(n, n − k), the unsigned Stirling number of the first kind. The bivariate exponential g.f. for sn,k(p) is given by ∑ n≥0 ∑ k≥0 x n n! q k = (1 − xq) … view at source ↗
Figure 5
Figure 5. Figure 5: The steps of implementing the mapping f in the proof of Theorem 6.1 showing that f(23471856) = 23457816. • In the case when xiri is an occurrence of but not of , there must be a unique element ai ∈ Si such that a1 ≤ ai < xi . After modification, xi becomes the smallest element in Si , a1ri is an occurrence of but not of , and there is no occurrence of involving ri because of the element ai+1. Now, if xjrj … view at source ↗
Figure 6
Figure 6. Figure 6: The steps in implementing the mapping f in the proof of Theorem 6.2, showing that f(24867315) = 26817345. The middle two diagrams represent the same permutation, and the rectangles indicate the regions in which the elements are moved by the cyclic shifts. The transformation f : Sn → Sn is obtained by considering ri , for i = ℓ, ℓ − 1, . . . , 1, one at a time. If ri , is not involved in an occurrence of ei… view at source ↗
Figure 7
Figure 7. Figure 7: Related to the proofs of Theorems 6.3 and 6.4. Theorem 6.3. The patterns in Class 73 are equivalent. For any pattern p in the class, sn,k(p) = ∑n−1 i=1 [i · sn−1,k−1,i(p) + (n − i) · sn−1,k,i(p)] (9) with the initial conditions s2,1(p) = 1, sn,0(p) = 1 for all n ∈ N, and s0,k(p) = 0 for all k ≥ 1, where sn,k,ℓ(p) is the number of n-permutations with k occurrences of p and ℓ right-to-left maxima satisfying … view at source ↗
Figure 8
Figure 8. Figure 8: The permutation π = 8297(10)64315 goes to σ = 7198(10)63425 under the bijection in the proof of Theorem 6.3. is no shading in squares (0,0), (0,2), (1,0), (1,2), and (2,0) in both patterns, the blocks Aj are independent in the sense that reversing the elements in any block cannot affect (create or destroy an occurrence of either of the patterns) another block. But then, we can consider each block independe… view at source ↗
Figure 9
Figure 9. Figure 9: The permutation π = 23784516 is mapped to f(π) = 74382156 under the bijection in the proof of Theorem 6.4. The rectangles indicate the areas where comple￾mentation occurs. 74382156 = f(π). Note that the occurrences 28, 38, 78, 46, 56 of in π are replaced by the occurrences 78, 48, 38, 26, 16 of in f(π). Similarly, the occurrences 28 and 16 of in π are replaced by the occurrences 78 and 58 of in f(π). We no… view at source ↗
Figure 10
Figure 10. Figure 10: Related to the proof of Theorem 7.1. where Hn = ∑n k=1 1 k is the n-th Harmonic number. Proof. We enumerate permutations avoiding the pattern in Class 47. We claim that F(x) = A(x) + A(x) ∑ n≥2 (n − 1)!Hn−1x n . (11) Indeed, each permutation (counted by F(x) on the LHS) either avoids the pattern (and hence is counted by A(x)) or contains it. Among all occurrences ab of the pattern in π, select the one wit… view at source ↗
Figure 11
Figure 11. Figure 11: Related to the proof of Theorem 7.2. Here, [x i ] denotes the coefficient of x i in the subsequent generating function, and c(n, k) is the unsigned Stirling number of the first kind, counting permutations of length n with k right-to-left maxima. We compute the number of n-permutations containing p and then subtract this from n!. Suppose that ab is an occurrence of p in π ∈ Sn, as shown in the left-hand di… view at source ↗
read the original abstract

We study the equidistribution of mesh patterns of length 2. We show that the number of equidistribution equivalence classes lies between 105 and 108, and conjecture that it is exactly 105. As a consequence, we obtain an upper bound of 49 Wilf-classes, improving the previously known bound of 56 due to Hilmarsson et al., and reducing the problem to three remaining conjectural equivalences (with the actual number conjectured to be 46). Our approach combines bijective constructions, generating functions, recurrence relations, and structural symmetries. We establish several new equidistribution results, including four previously unknown distribution classes, connect numerous patterns to known distributions in the literature, and resolve seven open pattern-avoidance enumeration problems posed by Hilmarsson et al. This work provides a near-complete classification of mesh patterns of length 2 and unifies several previously isolated results within a coherent framework.

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 / 2 minor

Summary. The paper examines the equidistribution of mesh patterns of length 2. It establishes that the number of equidistribution equivalence classes is between 105 and 108, with a conjecture that the exact number is 105. This yields an upper bound of 49 on the number of Wilf-classes, improving the previous bound of 56. The proofs utilize bijective constructions, generating functions, recurrence relations, and structural symmetries, while also resolving seven open enumeration problems from prior literature.

Significance. If the stated bounds and explicit results hold, this manuscript makes a substantial contribution to the study of mesh patterns by providing a near-complete classification for length-2 cases. The improvement in the Wilf-class bound and the unification of isolated results are notable. The resolution of multiple open problems enhances the practical value of the work.

minor comments (2)
  1. The abstract states that four previously unknown distribution classes are established, but does not specify which patterns they involve; listing them briefly would aid readers.
  2. The classification table would benefit from an additional column indicating the specific method (bijection, GF, etc.) used to establish each equivalence or distinction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on equidistribution of mesh patterns of length 2 and for recommending minor revision. The report correctly notes the bounds of 105-108 classes (conjectured 105), the resulting Wilf-class upper bound of 49, and the resolution of seven open enumeration problems. No specific major comments appear in the provided report.

Circularity Check

0 steps flagged

No significant circularity; bounds derived from independent bijections and generating functions

full rationale

The paper establishes the 105–108 range for equidistribution classes of length-2 mesh patterns via explicit bijective constructions, generating-function identities, recurrence relations, and structural symmetries that are developed directly in the work. These tools are applied to the 108 candidate patterns to prove equivalences and distinctions without reducing any central count to a fitted parameter or to a self-citation chain. Prior results from Hilmarsson et al. are cited only for context and for the baseline bound of 56; the improvements to 49 Wilf-classes and the resolution of seven open enumeration problems rest on the paper’s own proofs rather than on any imported uniqueness theorem or ansatz. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The classification rests on the standard combinatorial definitions of mesh patterns, equidistribution, and Wilf equivalence together with the correctness of the bijective and generating-function constructions; no new free parameters or postulated entities are introduced.

axioms (2)
  • standard math Mesh patterns of length 2 are defined by the usual placement rules on two distinguished entries together with the mesh constraints.
    Invoked throughout the classification to enumerate and compare patterns.
  • domain assumption Equidistribution is preserved under the bijective constructions and generating-function identities employed.
    Central to grouping patterns into equivalence classes.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages · 1 internal anchor

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