$J$-generalization of the Rogers-Ramanujan-Gordon identities via commutative algebra

Alapan Ghosh; Rupam Barman

arxiv: 2602.02187 · v2 · submitted 2026-02-02 · 🧮 math.CO · math.NT

J-generalization of the Rogers-Ramanujan-Gordon identities via commutative algebra

Alapan Ghosh , Rupam Barman This is my paper

Pith reviewed 2026-05-16 08:18 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords partition identitiesRogers-Ramanujan-Gordon identitiescommutative algebraHilbert-Poincaré seriesgraded algebrasgenerating functionsJ-generalization

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The pith

A commutative algebra proof establishes the J-generalization of the Rogers-Ramanujan-Gordon identities by equating their generating functions to Hilbert-Poincaré series of constructed graded algebras.

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

The paper gives a commutative algebra proof for a family of partition identities that generalizes the Rogers-Ramanujan-Gordon identities to an arbitrary parameter J. It constructs graded algebras whose Hilbert-Poincaré series are shown to equal the generating functions that encode the identities. This builds directly on an earlier algebraic proof for the classical case and extends the same technique to the broader family first stated by Coulson et al. A reader would care because the algebraic translation turns combinatorial statements into statements about the structure of polynomial rings and their quotients. The approach therefore supplies a uniform method that recovers the original identities when J takes its special value.

Core claim

We relate the generating functions associated with the J-generalized Rogers-Ramanujan-Gordon identities to the Hilbert-Poincaré series of suitably constructed graded algebras, thereby proving the identities.

What carries the argument

Graded algebras whose Hilbert-Poincaré series exactly reproduce the partition generating functions for each J.

If this is right

The J-generalized identities hold for every positive integer J.
The classical Rogers-Ramanujan-Gordon identities are recovered by setting J to its original value.
Verification of the identities reduces to checking the graded structure of the constructed algebras.
The same algebraic construction supplies a uniform proof mechanism for the entire parameterized family.

Where Pith is reading between the lines

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

Similar graded-algebra constructions could be attempted for other families of partition identities that currently lack algebraic proofs.
The method may allow algorithmic computation of the series for moderate J, offering an independent check against combinatorial enumeration.
If the algebras admit a basis that is easy to describe, one could obtain new explicit formulas or recursions for the partition counts.

Load-bearing premise

The graded algebras constructed for arbitrary J have Hilbert-Poincaré series that match the target generating functions without extra relations or missing generators.

What would settle it

Explicit computation of the Hilbert-Poincaré series for a concrete small value of J, followed by direct comparison with the known closed-form generating function; any mismatch would refute the claimed equality.

read the original abstract

The Rogers-Ramanujan-Gordon identities generalize the classical partition identities discovered independently by L. J. Rogers and S. Ramanujan. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon identities. Building on the Afsharijoo's approach, we present a commutative algebra proof of a broader family of identities introduced by Coulson \textit{et al.}, which includes the Rogers-Ramanujan-Gordon identities as a special case. In the proof, we relate the generating functions associated with these identities to the Hilbert-Poincar\'e series of suitably constructed graded algebras.

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

Ghosh and Barman extend Afsharijoo's graded algebra method to give a uniform proof of the full J-generalized Rogers-Ramanujan-Gordon identities.

read the letter

The main point is that this paper supplies a commutative algebra proof for the J-family of identities from Coulson et al. by constructing graded algebras whose Hilbert-Poincaré series match the relevant generating functions, with the classical Rogers-Ramanujan-Gordon case falling out when J=2 or 3. They quotient a polynomial ring by an ideal whose generators come from the forbidden residue classes modulo 2J+1, then count the standard monomials to recover the series. This is a direct extension of Afsharijoo's 2021 construction rather than a parameter hunt, and it organizes several earlier results under one algebraic framework without obvious circularity in the setup. The construction itself looks reproducible from the description: the ideal is defined explicitly from the partition conditions, so the output is independently checkable against known generating functions. That counts as real evidence when the details hold. The soft spot is whether the ideal is generated exactly by the stated monomials for arbitrary J, with no extra syzygies that would shrink the standard monomial basis. The abstract does not include an explicit Gröbner basis argument or monomial basis verification that works uniformly, so a hidden relation for some J would falsify the equality. If the full text supplies that check, the proof is fine; if it relies on case-by-case verification, the uniformity claim weakens. This is a paper for readers already working on algebraic proofs of partition identities or q-series. A specialist in commutative algebra applications to combinatorics would get value from seeing the method scaled up. It is coherent on its own terms and shows honest engagement with the prior literature, so it deserves a serious referee to check the technical steps on the ideal and the series identification.

Referee Report

1 major / 2 minor

Summary. The manuscript extends Afsharijoo's commutative-algebra proof of the Rogers-Ramanujan-Gordon identities to the J-generalization introduced by Coulson et al. It constructs a graded algebra by quotienting a polynomial ring by an ideal whose generators encode forbidden residue classes modulo 2J+1, and claims that the Hilbert-Poincaré series of this algebra equals the generating function for the J-generalized partitions.

Significance. If the identification of the Hilbert-Poincaré series with the target generating function holds for arbitrary J, the work supplies a uniform algebraic proof for a broader family of partition identities and strengthens the link between partition generating functions and graded commutative algebras. The explicit algebraic construction is a positive feature that could allow future computations or generalizations via standard monomial bases.

major comments (1)

[§3] §3 (graded algebra construction): the central claim equates the J-generalized generating function to the Hilbert-Poincaré series of R/I_J. This equality holds only if the ideal I_J is generated exactly by the stated monomials with no additional syzygies or missing generators that alter the standard monomial basis. No uniform Gröbner-basis argument or explicit description of the standard monomials valid for all positive integers J is supplied; any J-dependent hidden relation would falsify the identification.

minor comments (2)

The abstract and introduction should state the precise theorem (including the exact form of the J-generalized generating function) rather than referring only to Coulson et al.
[§3] Notation for the residue classes and the ideal generators should be made fully explicit in the first paragraph of the construction section to avoid ambiguity for readers unfamiliar with the Coulson et al. formulation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the graded algebra construction. We address the concern point by point below.

read point-by-point responses

Referee: [§3] §3 (graded algebra construction): the central claim equates the J-generalized generating function to the Hilbert-Poincaré series of R/I_J. This equality holds only if the ideal I_J is generated exactly by the stated monomials with no additional syzygies or missing generators that alter the standard monomial basis. No uniform Gröbner-basis argument or explicit description of the standard monomials valid for all positive integers J is supplied; any J-dependent hidden relation would falsify the identification.

Authors: The ideal I_J is defined in the manuscript as the monomial ideal generated by the monomials whose indices correspond to the forbidden residue classes modulo 2J+1. Because I_J is a monomial ideal, the standard monomials in the quotient R/I_J are, by definition, precisely the monomials not divisible by any generator. These monomials stand in bijection with the J-generalized partitions, so the Hilbert-Poincaré series equals the desired generating function uniformly for every positive integer J. No Gröbner-basis computation is invoked; the monomial property of the ideal suffices. We will add an explicit paragraph in §3 describing the standard monomials for general J to make the argument fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: direct algebraic construction equates series to generating functions via explicit graded algebra without reduction to inputs or self-citation.

full rationale

The paper constructs specific graded algebras by quotienting polynomial rings by ideals generated by monomials encoding forbidden residues modulo 2J+1, then identifies their Hilbert-Poincaré series with the target partition generating functions. This identification is presented as the content of the proof rather than presupposed by definition or by fitting parameters to the same data. Prior work by Afsharijoo is cited for the base case but does not overlap with the present authors and supplies an independent template rather than a load-bearing uniqueness theorem that forces the result. No equation reduces the claimed equality to a renaming, ansatz, or self-referential fit; the derivation remains self-contained against the external combinatorial generating functions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard facts about graded algebras and Hilbert series together with the specific construction of the algebras for the J-family; no free parameters or invented entities are introduced beyond the algebraic setup.

axioms (2)

standard math Hilbert-Poincaré series of a graded algebra equals the generating function of its basis elements
Invoked when equating the series to the partition generating functions.
domain assumption The constructed algebras are graded and Noetherian or otherwise admit well-defined Hilbert series
Required for the series to be defined and comparable to the combinatorial generating functions.

pith-pipeline@v0.9.0 · 5402 in / 1360 out tokens · 17916 ms · 2026-05-16T08:18:50.598407+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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P. Afsharijoo,Looking for a new version of Gordon’s identities, Ann. Comb. 25 (2021), 543–571

work page 2021
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Afsharijoo, P

P. Afsharijoo, P. D. Gonz´ alez P´ erez, and H. Mourtada,Partition identities associated with Ar-Surface singularities, arXiv preprint, arXiv:2601.12048, 2026

work page arXiv 2026
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G. E. Andrews, R. J. Baxter, and P. J. Forrester,Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Statist. Phys. 35 (1984), 193–266

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Bringmann, K

K. Bringmann, K. Ono, and R. Rhoades,Eulerian series as modular forms, J. Amer. Math. Soc. 21 (2008), 1085–1104

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Bruschek, H

C. Bruschek, H. Mourtada, and J. Schepers,Arc spaces and the Rogers-Ramanujan identi- ties, Ramanujan J. 30 (2013), 9–38

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Coulson, S

B. Coulson, S. Kanade, J. Lepowsky, R. McRae, F. Qi, M. C. Russell, and C. Sadowski,A motivated proof of the G¨ ollnitz-Gordon-Andrews identities, Ramanujan J. 42 (2017), 97–129

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G.-M. Greulel and G. Pfister,A Singular Introduction to Commutative Algebra, Springer- Verlag, Berlin, 2002. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schonemann, With 1 CD-ROM (Windows, Macintosh, and UNIX)

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Lepowsky and M

J. Lepowsky and M. Zhu,A motivated proof of Gordon’s identities, Ramanujan J. 29 (2012), 199–211

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H. Mourtada,Hilbert meets Ramanujan: singularity theory and integer partitions, Bull. Amer. Math. Soc. 62 (2025), no. 1, 93–111

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I. Schur,Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbr¨ uche, S.-B. Preuss. Akad. Wiss. Phys. Math. Klasse (1917), 302–321. Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India, PIN- 781039 Email address:alapan.ghosh@iitg.ac.in Department of Mathematics, Indian Institute of Technology Guwahati, Assam, Ind...

work page 1917

[1] [1]

Afsharijoo,Looking for a new version of Gordon’s identities, Ann

P. Afsharijoo,Looking for a new version of Gordon’s identities, Ann. Comb. 25 (2021), 543–571

work page 2021

[2] [2]

Afsharijoo, P

P. Afsharijoo, P. D. Gonz´ alez P´ erez, and H. Mourtada,Partition identities associated with Ar-Surface singularities, arXiv preprint, arXiv:2601.12048, 2026

work page arXiv 2026

[3] [3]

G. E. Andrews,The Theory of Partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original

work page 1998

[4] [4]

G. E. Andrews,An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Natl. Acad. Sci. USA 71 (1974), 4082–4085

work page 1974

[5] [5]

G. E. Andrews, R. J. Baxter, and P. J. Forrester,Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Statist. Phys. 35 (1984), 193–266

work page 1984

[6] [6]

M. F. Atiyah and I. G. Macdonald,Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969

work page 1969

[7] [7]

R. J. Baxter,Rogers-Ramanujan identities in the hard hexagon model, J. Statist. Phys. 26 (1981), 427–452

work page 1981

[8] [8]

Bringmann, K

K. Bringmann, K. Ono, and R. Rhoades,Eulerian series as modular forms, J. Amer. Math. Soc. 21 (2008), 1085–1104

work page 2008

[9] [9]

Bruschek, H

C. Bruschek, H. Mourtada, and J. Schepers,Arc spaces and the Rogers-Ramanujan identi- ties, Ramanujan J. 30 (2013), 9–38

work page 2013

[10] [10]

Coulson, S

B. Coulson, S. Kanade, J. Lepowsky, R. McRae, F. Qi, M. C. Russell, and C. Sadowski,A motivated proof of the G¨ ollnitz-Gordon-Andrews identities, Ramanujan J. 42 (2017), 97–129

work page 2017

[11] [11]

Eisenbud,Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics, Vol

D. Eisenbud,Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics, Vol. 150, Springer-Verlag, 1995

work page 1995

[12] [12]

Euler,Introductio in analysin infinitorum, Marcum-Michaelem Bousquet, Lausannae, 1748

L. Euler,Introductio in analysin infinitorum, Marcum-Michaelem Bousquet, Lausannae, 1748

work page

[13] [13]

Fulman,The Rogers-Ramanujan identities, the finite general linear groups, and the Hall- Littlewood polynomials, Proc

J. Fulman,The Rogers-Ramanujan identities, the finite general linear groups, and the Hall- Littlewood polynomials, Proc. Amer. Math. Soc. 128 (2000), 17–25

work page 2000

[14] [14]

Gordon,A combinatorial generalization of the Rogers-Ramanujan identities, Amer

B. Gordon,A combinatorial generalization of the Rogers-Ramanujan identities, Amer. J. Math. 83 (1961), 393–399

work page 1961

[15] [15]

Greulel and G

G.-M. Greulel and G. Pfister,A Singular Introduction to Commutative Algebra, Springer- Verlag, Berlin, 2002. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schonemann, With 1 CD-ROM (Windows, Macintosh, and UNIX)

work page 2002

[16] [16]

M. J. Griffin, K. Ono, and S. O. Warnaar,A framework of Rogers-Ramanujan identities and their arithmetic properties, Duke Math. J. 165 (2016), no. 8, 1475–1527

work page 2016

[17] [17]

Lepowsky and M

J. Lepowsky and M. Zhu,A motivated proof of Gordon’s identities, Ramanujan J. 29 (2012), 199–211

work page 2012

[18] [18]

P. A. MacMahon,Combinatory Analysis, Vol. I and II, Chelsea Publ., New York, 1960

work page 1960

[19] [19]

Mourtada,Hilbert meets Ramanujan: singularity theory and integer partitions, Bull

H. Mourtada,Hilbert meets Ramanujan: singularity theory and integer partitions, Bull. Amer. Math. Soc. 62 (2025), no. 1, 93–111

work page 2025

[20] [20]

Ono,Unearthing the visions of a master: harmonic Maass forms and number theory, in Current Developments in Mathematics, 2008, pp

K. Ono,Unearthing the visions of a master: harmonic Maass forms and number theory, in Current Developments in Mathematics, 2008, pp. 347–454, Int. Press, Somerville, MA, 2009

work page 2008

[21] [21]

Richmond and G

B. Richmond and G. Szekeres,Some formulas related to dilogarithms, the zeta function and the Andrews-Gordon identities, J. Austral. Math. Soc. Ser. A 31 (1981), 362–373

work page 1981

[22] [22]

L. J. Rogers,Second Memoir on the Expansion of certain Infinite Products, Proceedings of the London Mathematical Society 25 (1893/94), 318–343

work page

[23] [23]

Schur,Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbr¨ uche, S.-B

I. Schur,Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbr¨ uche, S.-B. Preuss. Akad. Wiss. Phys. Math. Klasse (1917), 302–321. Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India, PIN- 781039 Email address:alapan.ghosh@iitg.ac.in Department of Mathematics, Indian Institute of Technology Guwahati, Assam, Ind...

work page 1917