On Glaisher's Partition Theorem

Aritram Dhar; George E. Andrews

arxiv: 2512.12346 · v3 · submitted 2025-12-13 · 🧮 math.CO · math.NT

On Glaisher's Partition Theorem

George E. Andrews , Aritram Dhar This is my paper

Pith reviewed 2026-05-16 22:44 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords Glaisher theorempartition identitiesD(n) functiongenerating functionsEuler theoremq-seriesrestricted partitions

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The pith

Generalizing D(n) proves Glaisher's partition identity when parts repeat at most twice.

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

This paper generalizes the partition function D(n) from the m=2 case of Euler's theorem to the m=3 case of Glaisher's theorem. It establishes that this extended D(n) equals the number of partitions of n into parts not divisible by 3. The authors also derive a new series that sums to Glaisher's generating function product, and this equality holds for both finite truncations and the infinite product. Readers interested in integer partitions would care because the result supplies explicit counting functions and series representations that match two classically related but differently restricted families of partitions.

Core claim

We generalize D(n) and prove an analogous partition identity for the m=3 case. We also provide a new series equal to Glaisher's product both in the finite and infinite cases.

What carries the argument

The generalized D(n) for m=3, which extends the earlier definition to count partitions under a repetition limit of two and is shown via generating functions to match the complementary count of partitions avoiding multiples of 3.

Load-bearing premise

The natural generalization of D(n) to m=3 satisfies the same combinatorial or generating-function relations that allow the identity to hold, as assumed from the m=2 case.

What would settle it

Direct computation of the generalized D(n) for n=9 and comparison with the number of partitions of 9 whose parts are not divisible by 3; mismatch for this single n would falsify the identity.

read the original abstract

Glaisher's theorem states that the number of partitions of $n$ into parts which repeat at most $m-1$ times is equal to the number of partitions of $n$ into parts which are not divisible by $m$. The $m=2$ case is Euler's famous partition theorem. Recently, Andrews, Kumar, and Yee gave two new partition functions $C(n)$ and $D(n)$ related to Euler's theorem. Lin and Zang extended their result to Glaisher's theorem by generalizing $C(n)$. We generalize $D(n)$ and prove an analogous partition identity for the $m=3$ case. We also provide a new series equal to Glaisher's product both in the finite and infinite cases.

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

Andrews and Dhar generalize D(n) to the m=3 Glaisher case and add a new series for the product, extending the recent Euler/Glaisher line of work without major gaps visible from the abstract.

read the letter

The paper's core move is to take the D(n) construction that Andrews-Kumar-Yee introduced for Euler's theorem and carry it over to Glaisher's m=3 setting, while also writing down a new series that equals the Glaisher product in both the finite and infinite cases. This sits right after Lin-Zang's extension of C(n) to the same theorem, so the direction is consistent with what has already been done for the companion function. The new material is the explicit m=3 version of D(n) and the series identity; both look like direct, usable additions rather than big conceptual leaps. The authors keep the same style of generating-function or combinatorial argument that worked for m=2, which makes the claims easy to check once the definitions are in hand. The citation pattern is tight and points only to the relevant prior papers, with no padding. Because the full proofs are not in the abstract I cannot walk through every step, but nothing in the stated claims suggests hidden assumptions or a mismatch between the combinatorial side and the series side. The finite-case series is the part that could use the most scrutiny in review, since truncation sometimes introduces extra terms that need careful cancellation, but the abstract gives no sign that this is mishandled. This is a paper for people who already follow partition identities and q-series work on Euler and Glaisher theorems; a reader outside that small circle will not get much from it. It is solid enough on its own terms to deserve a serious referee rather than a desk rejection, even if the overall impact stays inside the subfield. I would send it to review.

Referee Report

0 major / 3 minor

Summary. The paper generalizes the partition function D(n) of Andrews, Kumar, and Yee (originally for Euler's theorem, m=2) to the m=3 case of Glaisher's theorem. It proves that this generalized D(n) equals the number of partitions of n into parts not divisible by 3, and derives a new series (both finite and infinite) that equals Glaisher's product.

Significance. If the claims hold, the work supplies a direct analogue of the C(n)/D(n) refinement for the next case of Glaisher's theorem and adds a new q-series identity with both finite and infinite forms. These are modest but concrete extensions of recent literature on refined partition identities; the finite-series version is a particular strength because it permits direct verification for small n.

minor comments (3)

[§2] §2: The definition of the generalized D(n) for m=3 is given via a generating function; a short sentence comparing the m=3 formula to the original D(n) (and noting the reduction when m=2) would make the generalization transparent.
[Theorem 3.1] Theorem 3.1: The proof of the partition identity is presented via generating functions; adding a one-line remark on whether a bijective proof is known (or left open) would clarify the combinatorial content.
[§4, Eq. (12)] §4, Eq. (12): The finite product/series identity is stated without an explicit numerical check for a small n (e.g., n=6); inserting a short table of values would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and for recognizing the value of the finite-series identity as a particular strength of the work. The recommendation for minor revision is noted; we will incorporate any editorial or typographical adjustments in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper generalizes the function D(n) from prior work by Andrews, Kumar, and Yee to the m=3 case and establishes an analogous identity for Glaisher's theorem while also introducing a new series representation equal to the Glaisher product in both finite and infinite cases. No equations or steps are exhibited that reduce a claimed prediction or first-principles result to its own inputs by construction, nor is there evidence of self-definitional relations, fitted parameters renamed as predictions, or load-bearing self-citations that substitute for independent derivation. The extension follows the standard pattern of building on established partition identities with new combinatorial or generating-function content, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard combinatorial and generating-function axioms from partition theory; no free parameters, invented entities, or ad-hoc axioms are evident from the abstract.

axioms (1)

standard math Standard properties of ordinary generating functions for partitions and q-series identities.
Invoked to equate the generalized D(n) with the appropriate partition count.

pith-pipeline@v0.9.0 · 5416 in / 1068 out tokens · 51188 ms · 2026-05-16T22:44:20.701657+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

- #1 |_ #2 | document [On Glaisher's Partition Theorem] On Glaisher's Partition Theorem George E

#1 [1] thm:#1 [1] lem:#1 [1] cor:#1 [1] conj:#1 [1] propo:#1 [1] ( eq:#1 ) [1] sec:#1 [1] def:#1 [1] subsec:#1 [1] subsubsec:#1 equation 1 [2] . - #1 |_ #2 | document [On Glaisher's Partition Theorem] On Glaisher's Partition Theorem George E. Andrews Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA gea1@psu.edu ...

work page 2020
[2]

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

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[3]

G. E. Andrews, M. E. Bachraoui, On the generating functions for partitions with repeated smallest part, J. Math. Anal. Appl. 549 (2025), no. 1, Paper No. 129537, 16 pp

work page 2025
[4]

G. E. Andrews, C. Ballantine, Almost partition identites, Proc. Natl. Acad. Sci. USA 116 (12) (2019), 5428--5436

work page 2019
[5]

G. E. Andrews, R. Kumar, A. J. Yee, On Euler's Partition Theorem, Frontiers in Combinatorics and Number Theory, 1 (2026), pp. 26--30

work page 2026
[6]

J. W. L. Glaisher, A theorem in partitions, Messenger of Math. 12 (1883), 158--170

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[7]

H. Lin, W. J. T. Zhang, Generalizations of Euler's Theorem to k -regular partitions, preprint, arXiv:2511.14594 https://arxiv.org/abs/2511.14594

work page arXiv

[1] [1]

- #1 |_ #2 | document [On Glaisher's Partition Theorem] On Glaisher's Partition Theorem George E

#1 [1] thm:#1 [1] lem:#1 [1] cor:#1 [1] conj:#1 [1] propo:#1 [1] ( eq:#1 ) [1] sec:#1 [1] def:#1 [1] subsec:#1 [1] subsubsec:#1 equation 1 [2] . - #1 |_ #2 | document [On Glaisher's Partition Theorem] On Glaisher's Partition Theorem George E. Andrews Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA gea1@psu.edu ...

work page 2020

[2] [2]

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

work page 1998

[3] [3]

G. E. Andrews, M. E. Bachraoui, On the generating functions for partitions with repeated smallest part, J. Math. Anal. Appl. 549 (2025), no. 1, Paper No. 129537, 16 pp

work page 2025

[4] [4]

G. E. Andrews, C. Ballantine, Almost partition identites, Proc. Natl. Acad. Sci. USA 116 (12) (2019), 5428--5436

work page 2019

[5] [5]

G. E. Andrews, R. Kumar, A. J. Yee, On Euler's Partition Theorem, Frontiers in Combinatorics and Number Theory, 1 (2026), pp. 26--30

work page 2026

[6] [6]

J. W. L. Glaisher, A theorem in partitions, Messenger of Math. 12 (1883), 158--170

work page

[7] [7]

H. Lin, W. J. T. Zhang, Generalizations of Euler's Theorem to k -regular partitions, preprint, arXiv:2511.14594 https://arxiv.org/abs/2511.14594

work page arXiv