Two-color partitions with evens in one color

George E. Andrews; Mohamed El Bachraoui

arxiv: 2512.23391 · v2 · pith:MBDWXIZHnew · submitted 2025-12-29 · 🧮 math.NT

Two-color partitions with evens in one color

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

Pith reviewed 2026-05-25 07:07 UTC · model grok-4.3

classification 🧮 math.NT

keywords two-color partitionseven parts restrictionpartition identitiesgenerating functionsparity constraintscolored partitionsinteger partitions

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The pith

Formulas for counting two-colored partitions with evens only in blue produce new identities among ordinary partitions.

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

The paper defines sequences that count partitions into red and blue parts with the rule that every even part must be blue. It then imposes further restrictions based on the parity of parts and their assigned colors. Explicit formulas are obtained for the sizes of these sequences. The formulas are used to derive identities that equate numbers of partitions satisfying different conditions. A sympathetic reader cares because the work connects a colored variant of partitions to classical identities without additional machinery.

Core claim

We consider sequences counting integer partitions in two colors (red and blue) in which the even parts occur only in blue color. We focus on subsequences defined by constraints on the parity and color of the summands. We establish formulas for our sequences and deduce identities of integer partitions.

What carries the argument

Generating functions built from the two-color assignment and the restriction that even parts appear only in blue, together with parity constraints on the parts.

If this is right

The counting sequences admit closed-form expressions or product representations.
Equalities hold between the number of partitions under one set of color-parity rules and the number under another set.
The same generating-function approach yields identities for additional parity conditions on the colored parts.

Where Pith is reading between the lines

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

The identities might admit direct bijective proofs that avoid generating functions.
The restriction of evens to one color could be varied to other arithmetic progressions while preserving the method.
Small-n computational checks would quickly test whether the formulas match direct counts.

Load-bearing premise

The generating functions constructed from the color and parity constraints correctly enumerate the restricted partitions without overcounting or omission.

What would settle it

An explicit enumeration of all qualifying colored partitions of some integer n that differs from the number given by the derived formula.

read the original abstract

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 restricts evens to one color in two-color partitions and derives formulas and identities for constrained subsequences.

read the letter

The key point is that this paper takes two-color partitions and adds the rule that even parts must be blue, then counts subsequences with parity and color constraints and derives some identities from those counts. The new element is that restriction on evens. It leads to different generating functions than the usual two-color cases where evens can be in either color. The focus on subsequences defined by those constraints is a specific choice that appears not to be in the prior work mentioned. The paper does well at framing the problem in a way that allows standard generating function techniques to be applied. If the calculations are carried through correctly, the formulas should follow directly from the product representations of the allowed parts. The soft spot is the absence of the actual generating functions or proof steps in the available information. The abstract asserts the formulas but to assess soundness we need to see if the enumeration avoids overcounting when parts have color and parity rules. That is the central assumption that needs verification. For readers who work on colored partitions and q-series, this adds a small new case that might inspire similar restrictions. It is not a major shift but could be useful for someone building on Andrews' earlier results in the area. I would send this to peer review. The claim is modest and the method is standard, so a referee can evaluate it without too much effort.

Referee Report

1 major / 0 minor

Summary. The paper considers sequences counting integer partitions in two colors (red and blue) where even parts occur only in blue. It focuses on subsequences defined by constraints on the parity and color of the summands, claims to establish formulas for these sequences, and deduces identities of integer partitions.

Significance. If the formulas and identities are correctly derived from standard generating-function constructions for colored partitions, the work would contribute new enumerative results and identities in the area of restricted partitions, building on classical techniques in partition theory.

major comments (1)

[Abstract] Abstract: the central claim that formulas are established and identities deduced cannot be assessed because the provided text supplies no generating functions, explicit derivations, or verification steps for the sequences.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript on two-color partitions with evens restricted to one color. The single major comment concerns the abstract; we address it directly below. The full manuscript contains the generating-function derivations and identities as claimed.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that formulas are established and identities deduced cannot be assessed because the provided text supplies no generating functions, explicit derivations, or verification steps for the sequences.

Authors: The abstract is a concise summary of the paper's main results. The full manuscript derives the generating functions for the two-color partitions (even parts only in blue) via standard product constructions over the allowed parts in each color and parity class. Explicit formulas are obtained for the subsequences under the stated parity/color constraints, and the partition identities are deduced by comparing coefficients or equating the generating functions to known series. These derivations and verifications appear in the body of the paper following the abstract. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs sequences for two-color partitions with even parts restricted to blue via standard generating functions from the stated color and parity constraints, then derives formulas and identities. No quoted steps reduce a claimed prediction or result to a fitted parameter, self-definition, or load-bearing self-citation chain; the approach begins from first-principles enumeration without the enumerated circular patterns. The central claims remain independent of their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.0 · 5555 in / 908 out tokens · 24875 ms · 2026-05-25T07:07:32.000281+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Arithmetic Properties of Overcolored Odd Partitions
math.NT 2026-05 unverdicted novelty 6.0

The paper proves new families of congruences modulo powers of 2 for the overcolored odd partition function bar a_s(n) for infinitely many s using generating functions, Hecke eigenforms, and Newman's results.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · cited by 1 Pith paper

[1]

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

work page 1998
[2]

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

work page 1987
[3]

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

work page 2021
[4]

G. E. Andrews and M. E. Bachraoui,On two-color partitions with odd smallest parts, preprint, https://arxiv.org/pdf/2410.14190, 2024

work page arXiv 2024
[5]

G. E. Andrews and R. Kumar,Rank, two-color partitions and mock theta functions, Proc. Amer. Math. Soc. 153 (2025), 4669–4682

work page 2025
[6]

G. E. Andrews and D. Newman,The minimal excludant in integer partitions, J. Integer Sequences 23 (2020), Article 20.2.3

work page 2020
[7]

Chen,On the number of overpartitions into odd parts, Discrete Math

S.-C. Chen,On the number of overpartitions into odd parts, Discrete Math. 325 (2014), 32–37

work page 2014
[8]

Corteel and J

S. Corteel and J. Lovejoy,Overpartitions, Trans. Amer. Math. Soc. 356 (2004), 1623–1635

work page 2004
[9]

Gasper and M

G. Gasper and M. Rahman,Basic Hypergeometric series, Cambridge University Press, 2004

work page 2004
[10]

M. D. Hirschhorn and J. A. Sellers,Arithmetic properties of overpartitions into odd parts, Ann. Comb. 10 (2006), 353–367. 10 G. E. ANDREWS AND M. EL BACHRAOUI The Pennsylvania State University, University Park, Pennsylvania 16802 Email address:andrews@math.psu.edu Dept. Math. Sci, United Arab Emirates University, PO Box 15551, Al-Ain, UAE Email address:me...

work page 2006

[1] [1]

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

work page 1998

[2] [2]

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

work page 1987

[3] [3]

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

work page 2021

[4] [4]

G. E. Andrews and M. E. Bachraoui,On two-color partitions with odd smallest parts, preprint, https://arxiv.org/pdf/2410.14190, 2024

work page arXiv 2024

[5] [5]

G. E. Andrews and R. Kumar,Rank, two-color partitions and mock theta functions, Proc. Amer. Math. Soc. 153 (2025), 4669–4682

work page 2025

[6] [6]

G. E. Andrews and D. Newman,The minimal excludant in integer partitions, J. Integer Sequences 23 (2020), Article 20.2.3

work page 2020

[7] [7]

Chen,On the number of overpartitions into odd parts, Discrete Math

S.-C. Chen,On the number of overpartitions into odd parts, Discrete Math. 325 (2014), 32–37

work page 2014

[8] [8]

Corteel and J

S. Corteel and J. Lovejoy,Overpartitions, Trans. Amer. Math. Soc. 356 (2004), 1623–1635

work page 2004

[9] [9]

Gasper and M

G. Gasper and M. Rahman,Basic Hypergeometric series, Cambridge University Press, 2004

work page 2004

[10] [10]

M. D. Hirschhorn and J. A. Sellers,Arithmetic properties of overpartitions into odd parts, Ann. Comb. 10 (2006), 353–367. 10 G. E. ANDREWS AND M. EL BACHRAOUI The Pennsylvania State University, University Park, Pennsylvania 16802 Email address:andrews@math.psu.edu Dept. Math. Sci, United Arab Emirates University, PO Box 15551, Al-Ain, UAE Email address:me...

work page 2006