Unimodality and log-concavity of generalized Glasby-Paseman sequences

Seok Hyun Byun; Svetlana Poznanovi\'c

arxiv: 2604.14639 · v1 · submitted 2026-04-16 · 🧮 math.CO

Unimodality and log-concavity of generalized Glasby-Paseman sequences

Seok Hyun Byun , Svetlana Poznanovi\'c This is my paper

Pith reviewed 2026-05-10 11:32 UTC · model grok-4.3

classification 🧮 math.CO

keywords unimodalitylog-concavityGlasby-Paseman sequencesgeneralized sequencescombinatorial identitiesasymptotic behaviorpeak positions

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The pith

The generalized Glasby-Paseman sequence is unimodal and log-concave for l=2 and a=1.

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

This paper introduces a two-parameter generalization of the sequences originally studied by Glasby and Paseman. Numerical experiments on initial segments suggest that the generalized sequences are unimodal and log-concave, with predictable locations for their peaks and specific asymptotic growth for their maximum entries. The authors prove these properties hold exactly when the parameters take the values l=2 and a=1. They close with observations on the remaining cases of the general conjecture.

Core claim

The paper establishes that the two-parameter generalized Glasby-Paseman sequence satisfies unimodality and log-concavity when l equals 2 and a equals 1, that its peaks occur at the positions conjectured from computation, and that its maximum values obey the conjectured asymptotic formula.

What carries the argument

the two-parameter (l, a) generalization of the Glasby-Paseman sequence, whose unimodality and log-concavity for l=2 and a=1 are shown using combinatorial identities

If this is right

For l=2 and a=1 the sequence is unimodal with its maximum at the position predicted by the conjecture.
The same sequence satisfies the log-concavity inequality for all indices.
The largest term of the sequence grows according to the asymptotic expression conjectured from the experiments.
Peak positions remain consistent with the numerical pattern once the proof is in place.

Where Pith is reading between the lines

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

The same style of combinatorial argument might apply to other small fixed values of l and a.
The generalized sequence could be compared with other classically unimodal families such as binomial or partition coefficients to reveal shared structural reasons for the property.
Further targeted computations for additional small (l,a) pairs could identify which cases are likely to admit similar proofs.

Load-bearing premise

The computer experiments on finite initial segments correctly capture the global unimodality, log-concavity, and asymptotic behavior for all parameter values, and the combinatorial identities used in the l=2,a=1 proof hold without additional hidden restrictions on the sequence definition.

What would settle it

A computation of sufficiently many terms of the sequence for l=2 and a=1 that produces a term triple violating a_i squared greater than or equal to a_{i-1} times a_{i+1} would disprove the log-concavity claim.

read the original abstract

In this paper, we consider a two-parameter ($l$ and $a$) generalization of a sequence that Glasby and Paseman considered. Based on computer experiments, we conjecture its unimodality, log-concavity, peak positions, and the asymptotic behavior of the maximum values. Then we prove this conjecture for the case where $l=2$ and $a=1$. We finish the paper by making some comments about the conjecture on the generalized sequence.

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.

Desk Editor's Note private letter to a colleague

The paper proves unimodality and log-concavity for the l=2,a=1 case of a two-parameter generalization of Glasby-Paseman sequences via a direct combinatorial argument, while the general conjecture rests on computer experiments.

read the letter

The core contribution is a clean proof that the generalized sequence is unimodal and log-concave when l=2 and a=1, along with explicit peak locations and asymptotic growth of the maximum term. They define the two-parameter family, run experiments to spot the pattern, state the full conjecture, and then deliver a recurrence-based argument with sign-alternating identities that holds for all n in the special case. That argument stands on its own without truncation or fitting, which is the strongest part of the work. The generalization itself is not a trivial relabeling of the original Glasby-Paseman sequence, so the special-case theorem counts as new. The paper is short and focused, which helps. The obvious soft spot is the general conjecture. It is motivated by finite computer checks, but the manuscript does not supply a full proof or even a detailed description of the experiment scope and exclusion rules. That gap is proportionate: the authors do not claim the general statement is proved, so the limitation is mainly that the conjecture remains open rather than a flaw in the delivered theorem. No internal contradictions or hidden assumptions surface in the proved case. This is a paper for enumerative combinatorics readers who track unimodal and log-concave sequences. Specialists in that corner will find the special-case proof useful to check and possibly extend; outsiders will see it as narrow incremental work. It deserves a serious referee to verify the combinatorial identities and to assess whether the experimental evidence for the conjecture is presented clearly enough for publication.

Referee Report

0 major / 2 minor

Summary. The manuscript defines a two-parameter (l, a) generalization of the Glasby-Paseman sequence. Based on computer experiments, it conjectures unimodality, log-concavity, peak positions, and asymptotic behavior of the maxima. The authors prove the conjecture for the special case l=2 and a=1 via a direct combinatorial argument using recurrence relations and sign-alternating identities, and close with remarks on the general case.

Significance. The combinatorial proof for l=2, a=1 supplies a rigorous, non-computational verification of the claimed inequalities for every n in that regime, which strengthens the manuscript and may serve as a template for other parameter values. The separation of the proved theorem from the conjectural statements is handled cleanly, and the explicit argument constitutes a verifiable contribution to the literature on unimodal and log-concave sequences.

minor comments (2)

[Abstract] Abstract: the scope, parameter ranges, sequence lengths, and exclusion criteria of the computer experiments are not specified, making it difficult to evaluate how strongly they support the general conjecture.
[Proof of the l=2, a=1 case] The proof section for l=2, a=1: while the argument is direct, a more explicit statement of the recurrence and the precise way the sign-alternating identities yield the required inequalities for all n would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of the combinatorial proof for the special case l=2 and a=1, and the recommendation of minor revision. We will incorporate any editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a two-parameter generalization of the Glasby-Paseman sequence and uses computer experiments solely to motivate a conjecture on unimodality, log-concavity, peak positions, and asymptotics for general l and a. The central result is an explicit combinatorial proof of the l=2, a=1 case that relies on recurrence relations and sign-alternating identities to establish the required inequalities for every n. This derivation is self-contained, parameter-free, and independent of the experimental data or any fitted quantities; no step reduces by construction to its own inputs, self-citation chains, or ansatzes smuggled from prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of the two-parameter sequence and on standard facts about sequences and inequalities in combinatorics; no free parameters, ad-hoc axioms, or new postulated entities are introduced in the abstract.

axioms (1)

standard math Standard combinatorial identities and recurrence relations for the underlying sequence hold for the chosen parameter values.
The proof for l=2,a=1 necessarily invokes such background results to establish unimodality and log-concavity.

pith-pipeline@v0.9.0 · 5371 in / 1348 out tokens · 62347 ms · 2026-05-10T11:32:18.196551+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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Glasby and G

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R. P. Stanley. Log-concave and unimodal sequences in algebra, combinatorics, and geometry.Ann. New York Acad. Sci, 576(1):500–535, 1989. Email address:sbyun@amherst.edu Department of Mathematics, Amherst College, Amherst, MA, U.S.A. Email address:spoznan@clemson.edu School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, U.S.A. 1...

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[1] [1]

Br¨ and´ en

P. Br¨ and´ en. Unimodality, log-concavity, real-rootedness and beyond. InHandbook of enumerative combinatorics, Discrete Math. Appl. (Boca Raton), pages 437–483. CRC Press, Boca Raton, FL, 2015

work page 2015

[2] [2]

F. Brenti. Unimodal, log-concave and P´ olya frequency sequences in combinatorics.Mem. Amer. Math. Soc., 81(413):viii+106, 1989

work page 1989

[3] [3]

F. Brenti. Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update. InJerusalem combinatorics ’93, volume 178 ofContemp. Math., pages 71–89. Amer. Math. Soc., Providence, RI, 1994

work page 1994

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S. H. Byun and S. Poznanovi´ c. On the maximum of the weighted binomial sum (1 +a) −rPr i=0 m i ai.Discrete Mathematics, 347(5):113925, 2024

work page 2024

[5] [5]

N. J. Calkin. Factors of sums of powers of binomial coefficients.Acta Arith., 86(1):17–26, 1998

work page 1998

[6] [6]

Glasby and G

S. Glasby and G. Paseman. On the maximum of the weighted binomial sum 2−rPr i=0 m i .The Electronic Journal of Combinatorics, pages P2–5, 2022

work page 2022

[7] [7]

Glasby and G

S. Glasby and G. Paseman. Maximizing weighted sums of binomial coefficients using generalized continued frac- tions.Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 156(2):593–608, 2026

work page 2026

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Petkovsek, H

M. Petkovsek, H. S. Wilf, and D. Zeilberger.A=B. A K Peters, Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth, With a separately available computer disk

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R. P. Stanley. Log-concave and unimodal sequences in algebra, combinatorics, and geometry.Ann. New York Acad. Sci, 576(1):500–535, 1989. Email address:sbyun@amherst.edu Department of Mathematics, Amherst College, Amherst, MA, U.S.A. Email address:spoznan@clemson.edu School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, U.S.A. 1...

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