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arxiv: 2306.16251 · v3 · pith:NKG6YSRInew · submitted 2023-06-28 · 🧮 math.CO · math.NT· math.RT

Companions to the Andrews-Gordon and Andrews-Bressoud Identities and Recent Conjectures of Capparelli, Meurman, Primc, and Primc

Matthew C. Russell This is my paper

Pith reviewed 2026-05-24 08:22 UTC · model grok-4.3

classification 🧮 math.CO math.NTmath.RT
keywords colored partitionsgenerating functionsAndrews-Gordon identitiesAndrews-Bressoud identitiescylindric partitionsbijectionspartition conjectures
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The pith

The k=1 cases of conjectured colored partition identities admit bivariate generating functions as slight variants of the Andrews-Gordon and Andrews-Bressoud sum sides.

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

The paper derives bivariate generating functions specifically for the k=1 instances of colored partition identities conjectured by Capparelli, Meurman, and the Primcs. These functions differ only slightly from the sum-side expressions in the Andrews-Gordon and Andrews-Bressoud identities. The resulting identities coincide with earlier results of Jing, Misra, and Savage. Explicit bijections are constructed using two-rowed cylindric partitions.

Core claim

We find bivariate generating functions for the k=1 cases of recently conjectured colored partition identities of Capparelli, Meurman, A. Primc, and M. Primc that are slight variants of the generating functions for the sum sides of the Andrews-Gordon and Andrews-Bressoud identities, relating them to recent work of Warnaar. These k=1 cases turn out to be equivalent to identities of Jing, Misra, and Savage. We provide bijections for these identities involving two-rowed cylindric partitions.

What carries the argument

Bivariate generating functions that are slight variants of the Andrews-Gordon and Andrews-Bressoud sum-side functions, together with bijections realized by two-rowed cylindric partitions.

If this is right

  • The k=1 cases satisfy the conjectured identities and are equivalent to the Jing-Misra-Savage identities.
  • The same slight variants of the Andrews-Gordon and Andrews-Bressoud generating functions count the colored partitions in these cases.
  • Explicit bijections exist between the two sides of each identity, realized by two-rowed cylindric partitions.
  • These results connect the conjectures to Warnaar's recent work on related generating functions.

Where Pith is reading between the lines

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

  • The pattern of slight variants may extend to higher k, suggesting a uniform generating-function approach for the full family of conjectures.
  • Cylindric-partition bijections could be adapted to prove other colored-partition identities that currently lack combinatorial proofs.
  • The equivalence to Jing-Misra-Savage identities opens the possibility of transferring representation-theoretic interpretations to the Capparelli-Meurman-Primc setting.

Load-bearing premise

The k=1 cases of the conjectured identities have generating functions exactly equal to the stated slight variants of the Andrews-Gordon and Andrews-Bressoud sum-side functions.

What would settle it

A mismatch between the coefficient of any monomial in the derived bivariate generating function and the actual count of the corresponding colored partitions for small values of the parameters would disprove the claimed equality.

read the original abstract

We find bivariate generating functions for the $k=1$ cases of recently conjectured colored partition identities of Capparelli, Meurman, A. Primc, and M. Primc that are slight variants of the generating functions for the sum sides of the Andrews-Gordon and Andrews-Bressoud identities, relating them to recent work of Warnaar. This $k=1$ cases turn out to be equivalent to identities of Jing, Misra, and Savage. Finally, we provide bijections for these identities involving two-rowed cylindric partitions, in the spirit of Corteel.

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.

Referee Report

0 major / 1 minor

Summary. The paper derives bivariate generating functions for the k=1 cases of the colored partition conjectures of Capparelli–Meurman–Primc–Primc; these functions are presented as explicit slight variants of the sum-side generating functions appearing in the Andrews–Gordon and Andrews–Bressoud identities. The k=1 cases are shown to be equivalent to identities of Jing–Misra–Savage, and the paper supplies explicit bijections realized by two-rowed cylindric partitions.

Significance. If the derivations hold, the work supplies concrete links between recent colored-partition conjectures and the classical Andrews–Gordon/Andrews–Bressoud framework, together with combinatorial proofs via cylindric partitions. The manuscript performs the generating-function derivations by direct q-series manipulation and constructs the bijections by explicit combinatorial mappings; these explicit, self-contained steps constitute a clear strength.

minor comments (1)
  1. Abstract, line beginning 'This k=1 cases turn out...': the sentence contains a subject–verb agreement error and should read 'These k=1 cases turn out...'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, including the accurate summary of our derivations of bivariate generating functions for the k=1 cases, the links to Andrews-Gordon/Andrews-Bressoud identities and Jing-Misra-Savage results, and the combinatorial bijections via two-rowed cylindric partitions. We note the recommendation for minor revision, but observe that the report contains no specific major comments requiring response or changes.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript derives the bivariate generating functions for the k=1 cases via explicit q-series manipulations that are slight variants of the Andrews-Gordon and Andrews-Bressoud sum sides, establishes equivalence to the Jing-Misra-Savage identities by direct comparison, and supplies explicit bijections through two-rowed cylindric partitions. All steps are carried out by standard generating-function identities and combinatorial mappings with no fitted parameters, self-definitional reductions, or load-bearing self-citations; the derivations remain independent of the input conjectures and are self-contained against external q-series and partition literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates entirely within standard combinatorial generating-function techniques and bijection constructions; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard algebraic properties of formal power series and partition generating functions hold.
    Invoked implicitly when equating the new bivariate functions to variants of Andrews-Gordon sum sides.

pith-pipeline@v0.9.0 · 5638 in / 1438 out tokens · 38059 ms · 2026-05-24T08:22:56.077575+00:00 · methodology

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Forward citations

Cited by 3 Pith papers

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

  1. Combinatorial construction of known positive series for partition classes defined by Capparelli, Meurman, Primc, and Primc in the $k$=1 Case

    math.CO 2026-05 unverdicted novelty 6.0

    The paper supplies a base-partition-and-moves combinatorial model for Russell's bivariate generating series of CMPP partitions in the k=1 case and completes several missing cases.

  2. Remarks on the conjectures of Capparelli, Meurman, Primc and Primc

    math.CO 2024-04 unverdicted novelty 6.0

    The authors relate the remaining two Capparelli-Meurman-Primc-Primc conjectures to non-standard specializations of standard modules for A_{2n}^{(2)} and D_{n+1}^{(2)} using prior Rogers-Ramanujan work for affine Lie algebras.

  3. Combinatorial construction of known positive series for partition classes defined by Capparelli, Meurman, Primc, and Primc in the $k$=1 Case

    math.CO 2026-05 unverdicted novelty 4.0

    The paper combinatorially constructs and extends known positive series for k=1 CMPP partitions via base partitions and moves, supplying missing cases from prior symbolic work.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · cited by 2 Pith papers

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