On a two-color partition series and its companions

George E. Andrews; Mohamed El Bachraoui

arxiv: 2606.30208 · v1 · pith:4IORDINBnew · submitted 2026-06-29 · 🧮 math.CO · math.NT

On a two-color partition series and its companions

George E. Andrews , Mohamed El Bachraoui This is my paper

Pith reviewed 2026-06-30 05:29 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords two-color partitionsdistinct partsRamanujan congruencesself-similarityoverpartitionsconcave compositionsquadratic forms

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The pith

The two-color distinct-part series S1(q) has its coefficients determined modulo 4 with a complete criterion for Ramanujan-type progressions, and its eta-normalized odd companion C(q) satisfies quintic self-similarity with coefficients nonze

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

The paper studies the two-color distinct-part generating function S1(q), equivalent to the generating function for strictly concave compositions. It determines the coefficients of S1(q) modulo 4 and gives a complete criterion for the Ramanujan-type progressions they satisfy. For the even companion Te(q), a direct overpartition interpretation is provided, showing that two partition families each account for half the coefficients. For the eta-normalized odd companion C(q) = (q;q)_∞ To(q), a quintic self-similarity is proved, along with exact vanishing relations, infinite sign changes in the coefficients, and the restriction that c(n) is nonzero only if 24n+28 is represented by x² + 3y².

Core claim

We determine the coefficients of S1(q) modulo 4 and obtain a complete criterion for the resulting Ramanujan-type progressions. For the even companion, we give a direct overpartition interpretation of its coefficients and show that two natural partition families are each counted by half of those coefficients. For the eta-normalized odd companion C(q)=(q;q)∞ To(q), we prove a quintic self-similarity, derive exact vanishing relations and infinite sign changes for its coefficients, and show that c(n) can be nonzero only when 24n+28 is represented by x²+3y².

What carries the argument

The quintic self-similarity of the eta-normalized odd companion C(q)=(q;q)_∞ To(q), which maps the series to a scaled version of itself and enforces the quadratic representation condition on the support of its coefficients.

If this is right

The coefficients of S1(q) can be completely determined modulo 4 in the relevant arithmetic progressions.
The coefficients of the even companion Te(q) are equally divided between two natural families of partitions or overpartitions.
The coefficients c(n) of C(q) vanish for all n not satisfying the representation condition by x² + 3y².
The coefficients of C(q) change sign infinitely often.
Exact relations hold for the vanishing of certain coefficients of C(q).

Where Pith is reading between the lines

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

The quintic self-similarity might permit an iterative description of the coefficient sequence or link to algebraic identities in q-series.
The restriction of support to quadratic forms of the type x² + 3y² could be used to estimate the density of nonzero coefficients.
The mod 4 analysis for S1(q) may generalize to other small moduli using similar dissection techniques.
This approach to companions could be applied to other generating functions in the theory of partitions to uncover similar self-similarities.

Load-bearing premise

The results depend on the equivalence of S1(q) to the generating function vd(q) for strictly concave compositions and on the given definitions of the odd and even companions.

What would settle it

A computation showing a nonzero coefficient c(n) of C(q) for some n where 24n+28 cannot be written as x² + 3y², or an S1(q) coefficient modulo 4 that violates the stated criterion for Ramanujan-type progressions.

read the original abstract

We study the two-color distinct-part series \(S_1(q)\), equivalently Andrews' generating function \(v_d(q)\) for strictly concave compositions, and its odd and even companions \(T_o(q)\) and \(T_e(q)\). We determine the coefficients of \(S_1(q)\) modulo \(4\) and obtain a complete criterion for the resulting Ramanujan-type progressions. For the even companion, we give a direct overpartition interpretation of its coefficients and show that two natural partition families are each counted by half of those coefficients. For the eta-normalized odd companion \(C(q)=(q;q)_\infty T_o(q)\), we prove a quintic self-similarity, derive exact vanishing relations and infinite sign changes for its coefficients, and show that \(c(n)\) can be nonzero only when \(24n+28\) is represented by \(x^2+3y^2\).

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

New mod-4 criteria and quintic self-similarity for the companions, but the opening equivalence to vd(q) is asserted without visible support.

read the letter

The paper gives a complete mod-4 criterion for the coefficients of S1(q) and spells out the resulting arithmetic progressions. For the even companion it supplies a direct overpartition interpretation plus a split into two natural families each accounting for half the count. For the eta-normalized odd companion it proves a quintic self-similarity, records exact vanishing rules, shows infinitely many sign changes, and ties nonzero coefficients to representation by x² + 3y². These are the concrete new statements.

The work sits inside the standard generating-function approach to restricted partitions and companions. Once the base series is fixed, the later steps look like routine q-series manipulations that can be checked directly.

The soft spot is the opening identification of S1(q) with Andrews' vd(q) for strictly concave compositions. The abstract states the equivalence but supplies no generating-function identity, coefficient comparison, or reference that would make it immediate. If the full text contains even a short derivation or a clear citation to a prior result that already equates the two, the rest of the claims attach without difficulty. If that step is missing or only implicit, the transfer of all coefficient results, interpretations, and the quadratic-form condition becomes the first thing a referee would ask to see clarified.

No free parameters or circular constructions appear. The claims are specific enough to be falsifiable by computation or direct expansion. This is the sort of targeted refinement that belongs in a partition-theory journal. A serious editor should send it to referees so the derivations can be verified and the equivalence settled.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the two-color distinct-part series S₁(q), which it identifies with Andrews' generating function v_d(q) for strictly concave compositions, together with its odd and even companions T_o(q) and T_e(q). It claims to determine the coefficients of S₁(q) modulo 4 and supply a complete criterion for the associated Ramanujan-type progressions; for the even companion it supplies a direct overpartition interpretation and shows that two natural partition families each account for half the coefficients; for the eta-normalized odd companion C(q)=(q;q)_∞ T_o(q) it proves a quintic self-similarity, exact vanishing relations, infinite sign changes in the coefficients, and the arithmetic condition that c(n) can be nonzero only when 24n+28 is represented by the form x²+3y².

Significance. If the central identification and the stated theorems are established with full proofs, the work would add concrete modular and arithmetic information to the study of two-color and companion partition series, including explicit links to quadratic forms and self-similar q-series structures.

major comments (1)

[Introduction] Introduction (opening paragraph): the equivalence S₁(q) ≡ v_d(q) is asserted without a generating-function identity, bijective proof, or coefficient comparison. Because every subsequent result on coefficients modulo 4, progressions, interpretations, self-similarity, vanishing, sign changes, and the x²+3y² condition rests on this identification, an explicit justification must be supplied.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for an explicit justification of the central identification. We address the single major comment below and will revise the manuscript to incorporate the requested material.

read point-by-point responses

Referee: [Introduction] Introduction (opening paragraph): the equivalence S₁(q) ≡ v_d(q) is asserted without a generating-function identity, bijective proof, or coefficient comparison. Because every subsequent result on coefficients modulo 4, progressions, interpretations, self-similarity, vanishing, sign changes, and the x²+3y² condition rests on this identification, an explicit justification must be supplied.

Authors: We agree that the equivalence S₁(q) = v_d(q) requires an explicit justification, as all subsequent theorems depend on it. In the revised manuscript we will insert a dedicated paragraph (or short subsection) immediately after the definition of S₁(q) that supplies the generating-function identity together with either a direct coefficient comparison or a bijective argument linking the two-color distinct parts to strictly concave compositions. This addition will be placed before any modular or arithmetic results are stated. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results derive from series analysis

full rationale

The paper identifies S1(q) equivalently with vd(q) from prior work by one author but derives all stated results (mod-4 coefficients, Ramanujan progressions, overpartition interpretations, quintic self-similarity of C(q), vanishing/sign-change relations, and x²+3y² representation condition) via direct generating-function manipulations and partition interpretations rather than by redefinition, fitting, or reduction to the identification itself. No equations or steps reduce claimed outputs to inputs by construction, and the cited equivalence is not invoked as a uniqueness theorem or load-bearing premise that forbids alternatives. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard q-series and partition theory; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)

standard math Standard definitions and algebraic properties of q-Pochhammer symbols and generating functions for partitions hold without further derivation.
Invoked implicitly when identifying S1(q) with vd(q) and defining the companions.

pith-pipeline@v0.9.1-grok · 5681 in / 1249 out tokens · 28159 ms · 2026-06-30T05:29:03.998980+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 5 canonical work pages

[1]

G. E. Andrews,The Theory of Partitions, Cambridge University Press, Cambridge, 1998

1998
[2]

G. E. Andrews,Rogers–Ramanujan identities for two-color partitions, Indian J. Math.29 (1987), 117–125

1987
[3]

G. E. Andrews,Partition identities for two-color partitions, Hardy–Ramanujan J.44(2021), 74–80

2021
[4]

G. E. Andrews,Concave and convex compositions, Ramanujan J.31(2013), no. 1, 67–82, doi:10.1007/s11139-012-9394-6

work page doi:10.1007/s11139-012-9394-6 2013
[5]

G. E. Andrews and M. El Bachraoui,Certain positiveq-series and inequalities for two-color partitions, Arab. J. Math. (2025).doi:10.1007/s40065-025-00567-3

work page doi:10.1007/s40065-025-00567-3 2025
[6]

G. E. Andrews and M. El Bachraoui,Congruences for two-color partitions with odd smallest part, preprint,arXiv:2410.14190 [math.NT], 2026

work page arXiv 2026
[7]

G. E. Andrews and B. C. Berndt,Ramanujan’s Lost Notebook, Part I, Springer, New York, 2005

2005
[8]

G. E. Andrews and R. Kumar,Rank, two-color partitions and mock theta functions, Proc. Amer. Math. Soc.153(2025), no. 11, 4669–4682

2025
[9]

Banerjee, K

K. Banerjee, K. Bringmann, and W. J. Keith,On a conjecture of Andrews and Bachraoui, preprint,arXiv:2601.04014 [math.NT], 2026

work page arXiv 2026
[10]

Chern and C

S. Chern and C. Wang,Proof of the Andrews–El Bachraoui positivity conjecture, preprint, arXiv:2601.19845 [math.CO], 2026

work page arXiv 2026
[11]

Corteel and J

S. Corteel and J. Lovejoy,Overpartitions, Trans. Amer. Math. Soc.356(2004), no. 4, 1623– 1635

2004
[12]

Gasper and M

G. Gasper and M. Rahman,Basic Hypergeometric Series, second ed., Encyclopedia of Math- ematics and its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004

2004
[13]

OEIS Foundation Inc.,Entry A344650 in The On-Line Encyclopedia of Integer Sequences, oeis.org/A344650(accessed April 16, 2026). The Pennsylvania State University, University Park, Pennsylvania 16802 Email address:andrews@math.psu.edu Department of Mathematical Sciences, United Arab Emirates University, PO Box 15551, Al-Ain, UAE Email address:melbachraoui@...

2026

[1] [1]

G. E. Andrews,The Theory of Partitions, Cambridge University Press, Cambridge, 1998

1998

[2] [2]

G. E. Andrews,Rogers–Ramanujan identities for two-color partitions, Indian J. Math.29 (1987), 117–125

1987

[3] [3]

G. E. Andrews,Partition identities for two-color partitions, Hardy–Ramanujan J.44(2021), 74–80

2021

[4] [4]

G. E. Andrews,Concave and convex compositions, Ramanujan J.31(2013), no. 1, 67–82, doi:10.1007/s11139-012-9394-6

work page doi:10.1007/s11139-012-9394-6 2013

[5] [5]

G. E. Andrews and M. El Bachraoui,Certain positiveq-series and inequalities for two-color partitions, Arab. J. Math. (2025).doi:10.1007/s40065-025-00567-3

work page doi:10.1007/s40065-025-00567-3 2025

[6] [6]

G. E. Andrews and M. El Bachraoui,Congruences for two-color partitions with odd smallest part, preprint,arXiv:2410.14190 [math.NT], 2026

work page arXiv 2026

[7] [7]

G. E. Andrews and B. C. Berndt,Ramanujan’s Lost Notebook, Part I, Springer, New York, 2005

2005

[8] [8]

G. E. Andrews and R. Kumar,Rank, two-color partitions and mock theta functions, Proc. Amer. Math. Soc.153(2025), no. 11, 4669–4682

2025

[9] [9]

Banerjee, K

K. Banerjee, K. Bringmann, and W. J. Keith,On a conjecture of Andrews and Bachraoui, preprint,arXiv:2601.04014 [math.NT], 2026

work page arXiv 2026

[10] [10]

Chern and C

S. Chern and C. Wang,Proof of the Andrews–El Bachraoui positivity conjecture, preprint, arXiv:2601.19845 [math.CO], 2026

work page arXiv 2026

[11] [11]

Corteel and J

S. Corteel and J. Lovejoy,Overpartitions, Trans. Amer. Math. Soc.356(2004), no. 4, 1623– 1635

2004

[12] [12]

Gasper and M

G. Gasper and M. Rahman,Basic Hypergeometric Series, second ed., Encyclopedia of Math- ematics and its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004

2004

[13] [13]

OEIS Foundation Inc.,Entry A344650 in The On-Line Encyclopedia of Integer Sequences, oeis.org/A344650(accessed April 16, 2026). The Pennsylvania State University, University Park, Pennsylvania 16802 Email address:andrews@math.psu.edu Department of Mathematical Sciences, United Arab Emirates University, PO Box 15551, Al-Ain, UAE Email address:melbachraoui@...

2026