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arxiv: 2510.07769 · v2 · submitted 2025-10-09 · 🧮 math.NT · math.CO

Strict Log-concavity of k-coloured Partitions

Kathrin Bringmann , Ben Kane , Anubhab Pahari , Larry Rolen This is my paper

Pith reviewed 2026-05-18 09:19 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords log-concavityk-coloured partitionsmajorizationpartition functionsstrict inequality
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The pith

The k-coloured partition function is strictly log-concave for every k at least 2.

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

This paper strengthens an earlier result by proving that the log-concavity of the k-coloured partition function p_k(n) is strict whenever k is two or more. The authors introduce the Hardy-Littlewood-Pólya majorization order on integer partitions of a fixed n and show that whenever one partition majorizes another, the value of p_k at the majorizing partition is strictly larger. This refines the previous proof of ordinary log-concavity that relied on recursive sequences and fractional-partition inequalities. A sympathetic reader would care because the strict version removes equality cases and therefore supplies sharper comparisons among these counting functions.

Core claim

For partitions a and b of the same natural number n, if b majorizes a, then p_k(b) exceeds p_k(a) whenever k is at least 2. This single inequality immediately yields strict log-concavity of the sequence p_k(n) for each such k. Numerical checks suggest the result cannot be improved further.

What carries the argument

The majorization partial order on partitions of a fixed integer, which ranks them according to the spread of their parts, applied to the generating function that enumerates k-coloured partitions.

If this is right

  • The sequence of k-coloured partition numbers is strictly log-concave for every k greater than or equal to 2.
  • Majorization gives a uniform way to compare p_k values across all partitions of n.
  • The strict inequality holds with explicit error bounds inherited from the earlier recursive construction.
  • Numerical evidence indicates that equality can occur only in the limiting or boundary cases already checked.

Where Pith is reading between the lines

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

  • The same majorization argument may apply to other weighted or restricted partition functions that admit similar recursions.
  • Strict comparisons of this type could tighten error terms in asymptotic formulas for coloured partition functions.
  • One could test whether the inequality persists when partitions are restricted to lie inside a fixed diagram or when colours are allowed to interact.

Load-bearing premise

The recursive sequences and fractional-partition inequalities from earlier work produce a strict increase under majorization with no equality cases once k reaches 2.

What would settle it

Two partitions a and b of the same n such that b majorizes a yet p_k(b) equals or is less than p_k(a) for some k at least 2.

read the original abstract

In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo--Pak, proved that the partition function $p(n)$ is eventually log-concave. Inspired by this and other results, Chern--Fu--Tang first conjectured log-concavity of $k$-coloured partitions. Three of the authors and Tripp later proved this conjecture by introducing recursive sequences and a strict inequality for fractional partition functions, giving explicit errors. In this paper, we show that the log-concavity is, in fact, strict for $k\geq 2$. We shed further light on this phenomenon by utilizing Hardy--Littlewood--P\'olya's notion of majorizing. We prove that for partitions $\bm{a},\bm{b}$ of $n\in\N$, if $\bm b$ majorizes $\bm a$, then $p_k(\bm{b})>p_k(\bm{a})$. Numerical calculations indicate that our result is sharp.

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

1 major / 2 minor

Summary. The manuscript proves strict log-concavity of the k-coloured partition function p_k for k≥2. Building on recursive sequences and strict fractional-partition inequalities from the authors' prior work with Tripp, it shows that if partitions a and b of the same n satisfy that b majorizes a in the Hardy–Littlewood–Pólya sense, then p_k(b) > p_k(a). Numerical checks are cited to indicate that the result is sharp.

Significance. If the central claim holds, the result strengthens existing log-concavity theorems for coloured partitions by replacing weak inequality with a strict combinatorial comparison via majorization. The approach supplies an explicit mechanism for the strictness and may facilitate analogous strengthenings for other partition statistics or generating functions.

major comments (1)
  1. [§3] §3, main theorem: the transfer of the strict inequality from the recursive sequences in the prior Tripp joint work to the majorization setting is asserted but not re-derived; an explicit one-paragraph recap of the relevant strict fractional-partition inequality (including the k≥2 case) would make the argument self-contained.
minor comments (2)
  1. [Notation] The boldface notation for partition vectors a, b is introduced without a preliminary definition; add a sentence in the notation subsection clarifying that these are integer partitions written in non-increasing order.
  2. [Table 1] Table 1 (numerical checks): the caption should state the range of n and k tested so that readers can assess the scope of the sharpness evidence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for the constructive suggestion to improve self-containment. We agree that a brief recap will strengthen the presentation and will incorporate the change in the revised version.

read point-by-point responses

  1. Referee: [§3] §3, main theorem: the transfer of the strict inequality from the recursive sequences in the prior Tripp joint work to the majorization setting is asserted but not re-derived; an explicit one-paragraph recap of the relevant strict fractional-partition inequality (including the k≥2 case) would make the argument self-contained.

    Authors: We agree that an explicit recap would make the argument more self-contained. In the revised manuscript we will add a single paragraph that recalls the strict fractional-partition inequality established in our prior work with Tripp, states the result explicitly for k ≥ 2, and indicates how the strict inequality transfers to the majorization comparison used in the proof of the main theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation applies the external combinatorial notion of majorization to the k-coloured partition function p_k and strengthens a prior log-concavity result by establishing strict inequality for k≥2. The recursive sequences and fractional-partition inequalities are cited from earlier joint work, but the current argument does not reduce any claim to a self-referential definition, fitted input, or unverified self-citation chain within this paper; the majorization comparison supplies independent content and the overall structure remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard facts about partitions and majorization together with recursive sequences proved in earlier work by three of the present authors plus Tripp; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Majorization is a partial order on integer partitions that respects the generating function for k-coloured partitions
    Invoked when the authors state that b majorizes a implies p_k(b) > p_k(a)
  • domain assumption The recursive sequences and strict fractional inequalities from the prior paper hold for the coloured case
    These are cited as the foundation for the non-strict result that is now strengthened

pith-pipeline@v0.9.0 · 5709 in / 1396 out tokens · 47776 ms · 2026-05-18T09:19:44.168133+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    Arnold,Majorization and the Lorenz order: A brief introduction.Springer New York, (2012)

    B. Arnold,Majorization and the Lorenz order: A brief introduction.Springer New York, (2012)

    work page 2012

  2. [2]

    Bessenrodt and K

    C. Bessenrodt and K. Ono,Maximal multiplicative properties of partitions, Ann. Comb. 20(1) (2016), 59–64

    work page 2016

  3. [3]

    Bringmann, B

    K. Bringmann, B. Kane, L. Rolen, and Z. Tripp,Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser, Trans. Amer. Math. Soc. Ser. B8(2021), 615–634

    work page 2021

  4. [4]

    Chern, S

    S. Chern, S. Fu, and D. Tang,Some inequalities fork-colored partition functions, Ra- manujan J.46(2018), 713–725

    work page 2018

  5. [5]

    Dalton,The measurement of the inequality of incomes, The Economic Journal30 (1920), 348–61

    H. Dalton,The measurement of the inequality of incomes, The Economic Journal30 (1920), 348–61

    work page 1920

  6. [6]

    DeSalvo and I

    S. DeSalvo and I. Pak,Log-concavity of the partition function, Ramanujan J.38(1):61– 73, 2015

    work page 2015

  7. [7]

    Engel,Strong properties in partially ordered sets II, Discrete Math.48(1984), 187– 196

    K. Engel,Strong properties in partially ordered sets II, Discrete Math.48(1984), 187– 196

    work page 1984

  8. [8]

    GöttscheThe Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math

    L. GöttscheThe Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann.286(1990), 192–207

    work page 1990

  9. [9]

    Han,The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applicationsAnn

    G.-N. Han,The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applicationsAnn. Inst. Fourier (Grenoble),60(1):1–29 (2010)

    work page 2010

  10. [10]

    Hardy, J

    G. Hardy, J. Littlewood, and G. Pólya,Some simple inequalities satisfied by convex functions.Messenger of Mathematics58(1929), 145-152

    work page 1929

  11. [11]

    Hardy, J

    G. Hardy, J. Littlewood, and G. Pólya,Inequalities, Cambridge Mathematical Library (1952)

    work page 1952

  12. [12]

    Jenkinson,Balanced words and majorization, Discrete mathematics, algorithms and applications1(2009), 463-483

    O. Jenkinson,Balanced words and majorization, Discrete mathematics, algorithms and applications1(2009), 463-483

    work page 2009

  13. [13]

    Lorenz,Methods of measuring the concentration of wealth, Publications of the Amer- ican Statistical Association9, no

    M. Lorenz,Methods of measuring the concentration of wealth, Publications of the Amer- ican Statistical Association9, no. 70 (1905), 209–219. 12

    work page 1905

  14. [14]

    Muirhead,Some methods applicable to identities and inequalities of symmetric al- gebraic functions of n letters, Proceedings of the Edinburgh Mathematical Society21 (1902), 144–162

    R. Muirhead,Some methods applicable to identities and inequalities of symmetric al- gebraic functions of n letters, Proceedings of the Edinburgh Mathematical Society21 (1902), 144–162

    work page 1902

  15. [15]

    Nekrasov and A

    N. Nekrasov and A. Okounkov,Seiberg–Witten theory and random partitions, The unity of mathematics, Birkhäuser Boston (2006), 525–596

    work page 2006

  16. [16]

    Nicolas,Sur les entiersnpour lesquels il y a beaucoup de groupes abéliens d’ordre n, Annales de l’Institut Fourier28(4) (1978), 1–16

    J.-L. Nicolas,Sur les entiersnpour lesquels il y a beaucoup de groupes abéliens d’ordre n, Annales de l’Institut Fourier28(4) (1978), 1–16

    work page 1978

  17. [17]

    Sagan,Inductive and injective proofs of log concavity results, Discrete Math.68 (1988), 281–292

    B. Sagan,Inductive and injective proofs of log concavity results, Discrete Math.68 (1988), 281–292

    work page 1988

  18. [18]

    Schur,Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinan- tentheorie, Sitzungsber

    I. Schur,Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinan- tentheorie, Sitzungsber. Berlin. Math. Ges.22(1923), 9–20

    work page 1923

  19. [19]

    Wang and H

    J. Wang and H. Zhang,q-weighted log-concavity and q-product theorem on the normality of posets, Adv. in Appl. Math.41(2008), 395–406. Department of Mathematics and Computer Science, Division of Mathemat- ics, University of Cologne, Weyertal 86–90, 50931 Cologne, Germany Email address:kbringma@math.uni-koeln.de Department of Mathematics, The University of ...

    work page 2008