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arxiv: 2307.09894 · v2 · pith:LRJ7BWEKnew · submitted 2023-07-19 · 🧮 math.CO

Schur-Positivity of Short Chords in Matchings

Avichai Marmor This is my paper

Pith reviewed 2026-05-24 07:46 UTC · model grok-4.3

classification 🧮 math.CO
keywords Schur positivitymatchingsshort chordsBessel polynomialsKnuth equivalencecrossing numberpattern avoidance
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The pith

Matchings with a fixed number of unmatched vertices are Schur-positive when graded by the number of short chords.

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

The paper proves that the generating function for matchings with a fixed number of unmatched vertices, graded by short chords, expands with nonnegative coefficients in the Schur basis of symmetric functions. Two proofs are given: one applying a combinatorial criterion for Schur-positivity and another that is explicitly bijective. The coefficients are identified as Bessel polynomials, and a Knuth-style equivalence relation partitions the matchings so that each class corresponds to an irreducible representation. Refined versions of the result hold for matchings with a prescribed crossing number or a given number of intersecting chord pairs. The paper also characterizes exactly which single matchings m have the property that the set of m-avoiding matchings remains Schur-positive.

Core claim

The set of matchings with a fixed number of unmatched vertices is Schur-positive with respect to the set of short chords. The Schur expansion coefficients are given explicitly by Bessel polynomials. A Knuth-like equivalence relation on matchings is defined such that each equivalence class corresponds to an irreducible representation. Various refined families, including matchings with fixed crossing number and matchings with a fixed number of pairs of intersecting chords, are also Schur-positive. All matchings m such that the m-avoiding matchings form a Schur-positive set are characterized.

What carries the argument

The short-chord statistic on matchings together with a combinatorial criterion for Schur-positivity.

Load-bearing premise

The new combinatorial criterion for Schur-positivity applies directly to the short-chord statistic on matchings without additional verification that the criterion's hypotheses hold in this setting.

What would settle it

A concrete counterexample would be any specific collection of matchings with a fixed number of unmatched vertices whose generating function by short chords has a negative coefficient when expanded in the Schur basis.

Figures

Figures reproduced from arXiv: 2307.09894 by Avichai Marmor.

Figure 1
Figure 1. Figure 1: The matching m = {(1, 3), (2, 6), (4, 5)}. Note that cyclic short chords are not counted. That is, for every m ∈ MN we have N /∈ Short(m). This holds even where (1, N) ∈ m. Enumerative properties of short chords in perfect matchings were studied extensively. The dis￾tribution of the number of short chords in perfect matchings appears in Entry A079267 of the OEIS [26], and was analyzed for example by Killpa… view at source ↗
Figure 2
Figure 2. Figure 2: A SYT of shape λ = (4, 2, 2, 1). The entry in row i and column j of a tableau T ∈ SYT(λ) is denoted as Ti,j . In addition, we define rowi(T) := {Ti,j | 1 ≤ j ≤ λi} as the set of entries in the i-th row of T. For example, if we consider the SYT shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The SYT Tλ for λ = (4, 2, 2, 1). The power of these notions may be reflected by the following statement: Lemma 3.7. Let λ, µ ⊢ N be two partitions. Then: (1) Des(Tλ) = [N − 1] \ {λ ′ 1 , λ′ 1 + λ ′ 2 , . . . }. (2) If Des(T) = [N − 1] \ {λ ′ 1 , λ′ 1 + λ ′ 2 , . . . } for some T ∈ SYT(µ), then µ ≥ λ in the conjugate order. Furthermore, if µ = λ then T = Tλ. Proof. The first assertion is obvious, so let us … view at source ↗
Figure 4
Figure 4. Figure 4: Example of the bijection for m1 = {(1, 2), (3, 5), (4)} ∈ M5,1 During the reduction process of m1, the short chord (1, 2) is removed. Then, the vertices 3, 4, 5 remain, so Stable(m1) = {3, 4, 5}. Next, the stable vertices are renumbered to {1, 2, 3}, so core(m1) = {(1, 3), (2)}, as in Figure 4b. In addition, the only unstable chord of m1 is (1, 2), so row2(T(m1)) = {2}. Therefore, T(m1) is the tableau pres… view at source ↗
Figure 5
Figure 5. Figure 5: Example of the bijection for m2 = {(1, 7), (2, 10), (3, 6), (4, 5), (8, 9)} ∈ M10,0 during another reduction process by e ′ 1 , . . . , e′ k ′, and denote e ′ ℓ = (i ′ ℓ , j′ ℓ ) for i ′ ℓ < j′ ℓ . It suffices to prove that eℓ ∈ {e ′ 1 , . . . , e′ k ′} for all 1 ≤ ℓ ≤ k (i.e., every chord that is removed during the first reduction process is removed during the second process as well). Assume by contradict… view at source ↗
Figure 6
Figure 6. Figure 6: A ballot path p ∈ P10,4 The bijection between SYTs of two rows and ballot paths is direct: Associate T ∈ SYT(N −k, k) with the path p(T) ∈ PN,N−2k such that UP(p(T)) = row1(T) and DOWN(p(T)) = row2(T). For example, consider the tableau T ∈ SYT(7, 3) described in Figure 5b, with row2(T) = {5, 6, 9}. It is associated with the ballot path p := p(T) ∈ P10,4 presented in [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Schreier graph of the natural action of S4 on PM4 Avni [3] proved that this graph is bipartite when ignoring loops. Therefore, we can create a layered graph by partitioning the vertices (matchings) into layers based on their distance from m0. Specifically, a matching in the ℓ-th layer is at a distance of ℓ from m0. Furthermore, a matching in one layer is connected to matchings in the adjacent layers, but n… view at source ↗
read the original abstract

We prove that the set of matchings with a fixed number of unmatched vertices is Schur-positive with respect to the set of short chords. Two proofs are presented. The first proof applies a new combinatorial criterion for Schur-positivity, while the second is bijective. The coefficients in the Schur expansion are derived, and interpreted in terms of Bessel polynomials. We present a Knuth-like equivalence relation on matchings, and show that every equivalence class corresponds to an irreducible representation. We proceed to find various refined Schur-positive sets, including the set of matchings with a prescribed crossing number and the set of matchings with a given number of pairs of intersecting chords. Finally, we characterize all the matchings $m$ such that the set of matchings avoiding $m$ is Schur-positive.

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 / 4 minor

Summary. The manuscript proves that the generating function over matchings with a fixed number of unmatched vertices, graded by the number of short chords, expands with nonnegative coefficients in the Schur basis. Two independent proofs are given: one via a new combinatorial criterion for Schur-positivity and one bijective. The Schur coefficients are identified with (signed) Bessel polynomials. A Knuth-like equivalence on matchings is introduced whose classes are in bijection with irreducible representations; refined Schur-positive subsets (fixed crossing number, fixed number of intersecting pairs) are exhibited; and a characterization is given of those matchings m for which the avoidance class is Schur-positive.

Significance. The result supplies new families of Schur-positive sets inside the matching lattice and links them to the representation theory of the symmetric group via an explicit Knuth equivalence. The existence of both a criterion proof and an independent bijective proof, together with the closed-form coefficient interpretation via Bessel polynomials, constitutes a solid combinatorial contribution. The avoidance characterization may be useful for further pattern-avoidance questions in this setting.

minor comments (4)
  1. [§1] §1: the definition of a 'short chord' (a chord connecting consecutive unmatched vertices or an edge of length 1 in the linear representation) should appear in the first paragraph rather than being deferred to the notation subsection.
  2. [Theorem 3.2] Theorem 3.2 (bijective proof): the construction of the bijection is stated only in outline; a short diagram or explicit description of how the short-chord statistic is preserved under the map would make the argument easier to verify.
  3. [§4.1] §4.1: the statement that each Knuth class 'corresponds to an irreducible representation' should be accompanied by an explicit reference to the Young diagram or Specht module that is realized.
  4. [throughout] The paper uses both 'short chord' and 'short edge' interchangeably in the later sections; a single consistent term would reduce minor confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the detailed summary, and the recommendation of minor revision. The report lists no specific major comments under the MAJOR COMMENTS section.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via independent bijective proof

full rationale

The paper states two independent proofs of the central Schur-positivity claim: one via a new combinatorial criterion for Schur-positivity and one bijective. The bijective proof does not rely on the criterion's hypotheses, making the main result independent of any unverified application of the new criterion. Coefficients, Knuth equivalence, refined sets, and avoidance characterization are presented as consequences rather than inputs. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the claim structure. The derivation chain remains self-contained against external combinatorial benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are visible. The work relies on standard background in symmetric functions and matchings.

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