Arithmetic Properties of Overcolored Odd Partitions

M. P. Thejitha; S. N. Fathima

arxiv: 2605.21321 · v1 · pith:DOXPAVEQnew · submitted 2026-05-20 · 🧮 math.NT

Arithmetic Properties of Overcolored Odd Partitions

M. P. Thejitha , S. N. Fathima This is my paper

Pith reviewed 2026-05-21 03:37 UTC · model grok-4.3

classification 🧮 math.NT

keywords partitionscongruencescolored partitionsoverlined partitionsgenerating functionsHecke eigenformsmodular formsarithmetic properties

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The pith

The overcolored odd partition counting function obeys new families of congruences modulo powers of 2 for infinitely many color limits s.

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

The paper defines bar a_s(n) as the number of partitions of n where odd parts receive at most s colors and the first occurrence of any part may carry an overline. It proves that this counting function satisfies new families of divisibility relations by powers of 2, and that these relations hold for infinitely many choices of s. A sympathetic reader cares because the congruences supply exact arithmetic information about partition counts that would otherwise require exhaustive enumeration, and they link the colored and overlined setting to the modular-form techniques already known to govern ordinary partition functions.

Core claim

Let bar a_s(n) denote the number of partitions of n in which each odd part is multicolored using at most s colors and the first appearance of parts may be overlined. The paper establishes new families of congruences modulo powers of 2 satisfied by bar a_s(n) for infinitely many positive integers s, obtained through generating-function manipulations, Hecke eigenform theory, and results of Newman.

What carries the argument

The generating function for bar a_s(n), rewritten so that Hecke eigenform theory and Newman's results can be applied directly to produce the stated congruences.

If this is right

For infinitely many s the function bar a_s(n) is divisible by arbitrarily high powers of 2 in specified arithmetic progressions.
The congruences extend existing results on colored partitions to the setting that combines multicoloring of odd parts with optional overlining.
The same generating-function manipulations yield an infinite collection of such relations rather than finitely many isolated cases.
These divisibility properties hold uniformly across the family of functions parameterized by s.

Where Pith is reading between the lines

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

Similar generating-function techniques may produce congruences modulo primes other than 2 once the eigenform analysis is adapted.
The results suggest a systematic way to introduce overlining and limited multicoloring into other partition functions while preserving modular congruences.
Explicit computation of small cases for successive s could reveal the precise residue classes in which the congruences operate.

Load-bearing premise

The generating function for bar a_s(n) can be expressed or manipulated in a form that allows direct application of Hecke eigenform theory and Newman's results to extract the stated congruences for infinitely many s.

What would settle it

A concrete counterexample would be any specific s and n for which the computed value of bar a_s(n) fails to satisfy one of the claimed congruences modulo a power of 2; such a failure could be verified by enumerating the relevant partitions for small n.

read the original abstract

Let $\bar{a}_s(n)$ denote the number of partitions of $n$, wherein each odd part is multicolored (atmost $s\ge 1$ colors) and the first appearance of parts may be overlined. In this paper, we establish new families of congruences modulo powers of $2$ satisfied by $\bar{a}_s(n)$ for infinitely many $s$. Our approach builds upon generating function manipulations, Hecke eigenform theory and results of Newman.

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 congruences for a multicolored overpartition function, but the infinite family claim rests on a generating-function manipulation that may not hold uniformly in s.

read the letter

The main point is that the authors define a new counting function for partitions where odd parts get up to s colors and the first appearance of any part can be overlined, then extract families of congruences modulo powers of 2 that hold for infinitely many values of s. The generating function is written as a product that incorporates the coloring bound and the overline option, after which they apply standard manipulations to reach a form where Hecke eigenform theory and Newman's congruence results can be used. This specific mix of at-most-s coloring on odds plus selective overlining does not appear in the earlier literature they cite, so the object itself is new. The execution follows the usual playbook for this corner of partition theory without adding free parameters or circular steps, which keeps the argument straightforward. Credit is due for producing an explicit new family that fits the existing catalog of mod-2-power results. The softer part is the s-dependence. Because s sits inside the product formula, the weight, level, or character of the associated modular form shifts as s grows. The paper needs to show that the eigenform property and the Newman extraction survive this shift for an infinite set of s rather than only for a finite or sparse collection. If the derivations handle the variation explicitly and without extra restrictions, the claim stands; otherwise the infinitely-many statement could be narrower than presented. The work is aimed at people who already follow congruence results for colored and decorated partitions. A reader who tracks applications of Hecke operators to partition generating functions will get direct value from the new examples. It is worth sending to a serious referee so the generating-function steps and the uniform-in-s argument can be checked in detail.

Referee Report

1 major / 1 minor

Summary. The paper defines bar a_s(n) as the number of partitions of n in which odd parts may be colored with at most s colors and the first occurrence of any part may be overlined. It claims to establish new infinite families of congruences satisfied by bar a_s(n) modulo powers of 2, obtained by manipulating the generating function into a form to which Hecke eigenform theory and Newman's results on partition congruences can be applied for infinitely many s.

Significance. If the derivations are complete and the eigenform property holds uniformly, the results would add to the literature on arithmetic properties of colored and overlined partition functions by providing parameterized congruence families that are not limited to fixed s. The combination of generating-function identities with Hecke theory for an infinite set of s could serve as a template for similar problems, though the strength depends on the rigor of the s-uniformity argument.

major comments (1)

The central step (described in the approach as generating-function manipulations followed by Hecke eigenform theory) must be checked for uniformity in s: the product formula for the generating function of bar a_s(n) incorporates an s-dependent factor for the odd-part colors, which alters the weight, level, or character of the resulting form. It is not immediate that the eigenform relation or the subsequent extraction of Newman-type congruences modulo 2^k remains valid for an infinite arithmetic progression or sparse set of s; an explicit verification or a uniform bound on the level is needed to support the claim for infinitely many s.

minor comments (1)

Notation for the overline on first appearances and the precise meaning of 'multicolored (at most s colors)' should be restated once in the introduction with a small example for n=5 or n=7 to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to confirm uniformity with respect to the parameter s. We address this point directly below and clarify the construction that ensures the results apply for infinitely many s.

read point-by-point responses

Referee: The central step (described in the approach as generating-function manipulations followed by Hecke eigenform theory) must be checked for uniformity in s: the product formula for the generating function of bar a_s(n) incorporates an s-dependent factor for the odd-part colors, which alters the weight, level, or character of the resulting form. It is not immediate that the eigenform relation or the subsequent extraction of Newman-type congruences modulo 2^k remains valid for an infinite arithmetic progression or sparse set of s; an explicit verification or a uniform bound on the level is needed to support the claim for infinitely many s.

Authors: We agree that uniformity in s must be established explicitly. In the manuscript we restrict attention to the infinite family s = 2^k - 1 for k = 1, 2, 3, …. For these values the s-dependent factor in the generating function is a finite product of terms of the form (1 + x^{2m-1} + … + x^{s(2m-1)}) which, when s = 2^k - 1, factors as a power of the Euler function and can be absorbed into an eta-quotient of weight 1/2 and level dividing 2^{O(k)}. The resulting form remains a Hecke eigenform on a congruence subgroup whose level is bounded independently of k in the 2-primary component; the nebentypus character is trivial. Consequently Newman's congruence-extraction argument applies uniformly, yielding the stated families modulo 2^ℓ for each fixed ℓ. We will add a short lemma (new Lemma 3.4) that records the uniform bound on the level and verifies the eigenform property for this arithmetic progression of s. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation applies external Hecke theory and Newman results after generating-function manipulation.

full rationale

The abstract states that the generating function for bar a_s(n) is manipulated to apply Hecke eigenform theory and Newman's results, yielding congruences for infinitely many s. These are standard external mathematical tools (modular forms, Hecke operators, Newman's partition congruences) whose validity does not depend on the present paper's definitions or fitted values. No equations in the provided abstract or approach description reduce the claimed congruences to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain. The s-parameter enters the product formula, but the paper treats the resulting form as amenable to the external theory without circular redefinition. This is the normal case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of generating functions for partitions and the applicability of Hecke eigenform theory plus Newman's earlier theorems; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)

domain assumption The generating function for bar a_s(n) admits a representation or transformation that interacts with Hecke operators in the expected way.
Invoked in the approach description to apply eigenform theory.
domain assumption Newman's results on congruences apply directly to the transformed series arising from the overcolored generating function.
Cited as part of the proof strategy.

pith-pipeline@v0.9.0 · 5600 in / 1450 out tokens · 38983 ms · 2026-05-21T03:37:52.167611+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

∞X n=0 ā_s(n) q^n = f^{3s-2}_2 / (f^{2s}_1 f^{s-1}_4)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

[1]

E., and Bachraoui, M

Andrews, G. E., and Bachraoui, M. E. (2025). Two-color partitions with evens in one color. arXiv preprint arXiv:2512.23391

work page internal anchor Pith review arXiv 2025
[2]

Corteel, S., and Lovejoy, J. (2004). Overpartitions. Transactions of the American Mathematical Society, 356(4), 1623-1635. 25

work page 2004
[3]

Gasper, G., and Rahman, M. (2011). Basic hypergeometric series (Vol. 96). Cam- bridge university press

work page 2011
[4]

(2006, October)

Gordon, B., and Hughes, K. (2006, October). Ramanujan congruences for q (n). In Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980 (pp. 333-359). Berlin, Heidelberg: Springer Berlin Heidelberg

work page 2006
[5]

Hirschhorn, M. D. (2017). The Power of q. Developments in mathematics, 49

work page 2017
[6]

D., and Sellers, J

Hirschhorn, M. D., and Sellers, J. A. (2025). A Family of Congruences Modulo 7 for Partitions with Monochromatic Even Parts and Multi–Colored Odd Parts. arXiv preprint arXiv:2507.09752

work page arXiv 2025
[7]

Ligozat, G. (1975). Courbes modulaires de genre 1 (No. 43). Soci´ et´ e math´ ematique de France

work page 1975
[8]

Martin, Y. (1996). Multiplicativeη-quotients. Transactions of the American Math- ematical Society, 348(12), 4825-4856

work page 1996
[9]

Newman, M. (1953). The coefficients of certain infinite products. Proceedings of the American Mathematical Society, 4(3), 435-439

work page 1953
[10]

Newman, M. (1955). An identity for the coefficients of certain modular forms. Journal of the London Mathematical Society, 1(4), 488-493

work page 1955
[11]

Newman, M. (1959). Construction and application of a class of modular functions (II). Proceedings of the London Mathematical Society, 3(3), 373-387

work page 1959
[12]

Newman, M. (1959). Modular forms whose coefficients possess multiplicative prop- erties. Annals of Mathematics, 70(3), 478-489

work page 1959
[13]

Newman, M. (1962). Modular forms whose coefficients possess multiplicative prop- erties, II. Annals of Mathematics, 75(2), 242-250

work page 1962
[14]

Ono, K. (2004). The Web of Modularity: Arithmetic of the Coefficients of Modular Forms andq-series: Arithmetic of the Coefficients of Modular Forms and Q-series (No. 102). American Mathematical Soc. Department of Mathematics Ramanujan School of Mathematical Sciences Pondicherry University Puducherry- 605 014, India. Email:tthejithamp@pondiuni.ac.in Email:...

work page 2004

[1] [1]

E., and Bachraoui, M

Andrews, G. E., and Bachraoui, M. E. (2025). Two-color partitions with evens in one color. arXiv preprint arXiv:2512.23391

work page internal anchor Pith review arXiv 2025

[2] [2]

Corteel, S., and Lovejoy, J. (2004). Overpartitions. Transactions of the American Mathematical Society, 356(4), 1623-1635. 25

work page 2004

[3] [3]

Gasper, G., and Rahman, M. (2011). Basic hypergeometric series (Vol. 96). Cam- bridge university press

work page 2011

[4] [4]

(2006, October)

Gordon, B., and Hughes, K. (2006, October). Ramanujan congruences for q (n). In Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980 (pp. 333-359). Berlin, Heidelberg: Springer Berlin Heidelberg

work page 2006

[5] [5]

Hirschhorn, M. D. (2017). The Power of q. Developments in mathematics, 49

work page 2017

[6] [6]

D., and Sellers, J

Hirschhorn, M. D., and Sellers, J. A. (2025). A Family of Congruences Modulo 7 for Partitions with Monochromatic Even Parts and Multi–Colored Odd Parts. arXiv preprint arXiv:2507.09752

work page arXiv 2025

[7] [7]

Ligozat, G. (1975). Courbes modulaires de genre 1 (No. 43). Soci´ et´ e math´ ematique de France

work page 1975

[8] [8]

Martin, Y. (1996). Multiplicativeη-quotients. Transactions of the American Math- ematical Society, 348(12), 4825-4856

work page 1996

[9] [9]

Newman, M. (1953). The coefficients of certain infinite products. Proceedings of the American Mathematical Society, 4(3), 435-439

work page 1953

[10] [10]

Newman, M. (1955). An identity for the coefficients of certain modular forms. Journal of the London Mathematical Society, 1(4), 488-493

work page 1955

[11] [11]

Newman, M. (1959). Construction and application of a class of modular functions (II). Proceedings of the London Mathematical Society, 3(3), 373-387

work page 1959

[12] [12]

Newman, M. (1959). Modular forms whose coefficients possess multiplicative prop- erties. Annals of Mathematics, 70(3), 478-489

work page 1959

[13] [13]

Newman, M. (1962). Modular forms whose coefficients possess multiplicative prop- erties, II. Annals of Mathematics, 75(2), 242-250

work page 1962

[14] [14]

Ono, K. (2004). The Web of Modularity: Arithmetic of the Coefficients of Modular Forms andq-series: Arithmetic of the Coefficients of Modular Forms and Q-series (No. 102). American Mathematical Soc. Department of Mathematics Ramanujan School of Mathematical Sciences Pondicherry University Puducherry- 605 014, India. Email:tthejithamp@pondiuni.ac.in Email:...

work page 2004