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arxiv: 2606.11011 · v1 · pith:JAGQ7JHXnew · submitted 2026-06-09 · 🧮 math.NT · math.CO

Weighted partitions with interval restrictions: exact formulas and a bivariate master identity

George E. Andrews , Mohamed El Bachraoui , Aritram Dhar , Ankush Goswami , Runqiao Li This is my paper

Pith reviewed 2026-06-27 11:30 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords partition functionsfalse theta seriesgenerating functionsinterval-restricted partitionssigned partitionsmaster identityRogers-Fine identity
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The pith

An auxiliary variable z tracking non-compulsory parts produces a master identity that yields the false theta formula for a2''(n) and restricts b2''(n) coefficients to -1,0,1,2.

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

The paper proves two conjectures on the signed partition functions a2''(n) and b2''(n) for interval-restricted partitions controlled by the smallest even part and the number of ones. It introduces an auxiliary variable z recording the number of non-compulsory parts greater than 1 to obtain closed forms for the bivariate generating functions A(z,q) and B(z,q). These satisfy the master identity (1+q^2)B(z,q)-(1+q)A(z,q)=-q^4/(1-q^3), proved both analytically and combinatorially. Specializing at z=-1 and applying a Rogers-Fine evaluation then gives the generating function for a2''(n) as a rational term plus false theta series, plus an explicit form for b2''(n) whose coefficients lie in the stated range.

Core claim

The central structural result introduces an auxiliary variable z recording the number of non-compulsory parts greater than 1. Closed forms are obtained for the two resulting generating functions A(z,q) and B(z,q), and the master identity (1+q^2)B(z,q)-(1+q)A(z,q)=-q^4/(1-q^3) is proved using both analytic and combinatorial techniques. At z=-1, this identity together with a Rogers-Fine evaluation gives the false theta formula for a2''(n) and an explicit generating function for b2''(n) that implies the coefficient range -1,0,1,2.

What carries the argument

The bivariate generating functions A(z,q) and B(z,q) with auxiliary variable z recording the number of non-compulsory parts greater than 1, which satisfy the stated master identity under the interval restrictions.

If this is right

  • The generating function for a2''(n) equals an elementary rational term plus a false theta series with periodic signs.
  • The coefficients b2''(n) take only the values -1,0,1,2 and admit an exact coefficient description.
  • Ordinary and fixed-refinement consequences follow directly from the master identity.
  • A quantum modular interpretation follows from the false theta series for a2''(n).
  • A direct Heine-Rogers-Fine proof of the false theta formula is obtained as a corollary.

Where Pith is reading between the lines

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

  • The bivariate technique with auxiliary z could be applied to other families of restricted partitions to derive similar master identities.
  • The explicit form for b2''(n) may allow closed-form expressions for its partial sums or moments.
  • The periodic signs in the false theta series suggest possible links to other periodic q-series identities beyond the Rogers-Fine setting.
  • Computational verification of the coefficient bounds for large n would provide independent confirmation before seeking a bijective proof.

Load-bearing premise

The generating functions A(z,q) and B(z,q) obtained after introducing the auxiliary variable z admit closed forms that satisfy the stated master identity under the given interval restrictions on the partitions.

What would settle it

Extract the explicit generating function for b2''(n) from the master identity at z=-1 and compute its coefficients for n up to several hundred; any coefficient outside {-1,0,1,2} would falsify the claimed range.

read the original abstract

Let $a_2''(n)$ and $b_2''(n)$ be the signed partition functions introduced by Andrews and El Bachraoui for interval-restricted partitions whose parts greater than $1$ are controlled by the smallest even part and by the number of ones. We prove two conjectures for these functions. The first gives the generating function for $a_2''(n)$ as an elementary rational term plus a false theta series with periodic signs; the second asserts that the companion coefficients $b_2''(n)$ take only the values $-1,0,1,2$. The central structural result introduces an auxiliary variable $z$ recording the number of non-compulsory parts greater than $1$. We obtain closed forms for the two resulting generating functions and prove the master identity $(1+q^2)\mathcal B(z,q)-(1+q)\mathcal A(z,q)=-q^4/(1-q^3)$ using both analytic and combinatorial techniques. At $z=-1$, this identity, together with a Rogers--Fine evaluation, gives the false theta formula for $a_2''(n)$ and an explicit generating function for $b_2''(n)$. The latter formula implies the asserted coefficient range and leads to an exact coefficient description of $b_2''(n)$. We also include a direct Heine--Rogers--Fine proof of the false theta formula, ordinary and fixed-refinement consequences of the master identity, and the resulting quantum modular interpretation.

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 / 2 minor

Summary. The paper proves two conjectures of Andrews and El Bachraoui on the signed partition functions a₂''(n) and b₂''(n) for interval-restricted partitions. It introduces an auxiliary variable z tracking non-compulsory parts >1, obtains closed forms for the bivariate generating functions A(z,q) and B(z,q), and establishes the master identity (1+q²)B(z,q)−(1+q)A(z,q)=−q⁴/(1−q³) via both analytic and combinatorial arguments. Specializing at z=−1 and applying a Rogers–Fine identity yields the false theta series for a₂''(n) and an explicit generating function for b₂''(n) whose coefficients lie in {−1,0,1,2}, together with an exact coefficient formula. A direct Heine–Rogers–Fine proof of the false theta formula, ordinary and refined consequences, and a quantum modular interpretation are also supplied.

Significance. If the derivations hold, the work resolves the two conjectures with explicit formulas and a coefficient bound, while the bivariate master identity supplies a new structural relation among the generating functions. The dual analytic/combinatorial proofs of the identity, the direct false-theta derivation, and the extraction of the coefficient range from the explicit series constitute concrete advances. The quantum-modular consequence situates the results in a broader q-series context.

minor comments (2)
  1. [§1] §1, paragraph after (1.3): the notation a₂''(n), b₂''(n) is introduced without an explicit reminder of the original definitions from Andrews–El Bachraoui; a one-sentence recap would aid readability.
  2. [§3] After Theorem 3.1: the combinatorial proof of the master identity is summarized in one paragraph; expanding the bijection or sign-reversing involution into two or three explicit steps would make the argument easier to follow without lengthening the paper substantially.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work, as well as for the recommendation to accept the manuscript. The report correctly identifies the resolution of the two conjectures, the bivariate master identity, and the additional consequences including the quantum-modular interpretation.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper first defines the bivariate generating functions A(z,q) and B(z,q) directly from the interval-restricted partition conditions with auxiliary variable z, then derives their closed forms. The master identity is proved independently via analytic manipulation of those closed forms and via a separate combinatorial argument; neither step reduces to the target false-theta or coefficient-range statements. Specialization at z=-1 plus the external Rogers-Fine identity (and a direct Heine-Rogers-Fine proof supplied in the paper) produces the claimed formulas. The only self-citation is to the prior introduction of a2''(n) and b2''(n) themselves; that citation supplies the objects of study but none of the load-bearing identities or evaluations. No fitted-input, self-definitional, or ansatz-smuggling steps appear. The chain is therefore externally verifiable and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on classical results in q-series and partition theory without introducing new parameters or entities.

axioms (2)
  • standard math Standard algebraic manipulations of generating functions are valid.
    Used throughout to derive the master identity from the partition definitions.
  • domain assumption The Rogers-Fine identity applies at the specified value of z.
    Invoked to obtain the false theta formula from the master identity.

pith-pipeline@v0.9.1-grok · 5818 in / 1273 out tokens · 36095 ms · 2026-06-27T11:30:36.236274+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references

  1. [1]

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

    1998

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    G. E. Andrews and B. C. Berndt,Ramanujan’s Lost Notebook, Part I, Springer, New York, 2005

    2005

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    G. E. Andrews and M. El Bachraoui,Formulas and conjectures for partitions with restrictions on interval of parts, Advances in Applied Mathematics 173 (2026), Paper No. 102981

    2026

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    G. E. Andrews, M. V. Subbarao, and M. Vidyasagar,A family of combinatorial identities, Canadian Mathematical Bulletin 15 (1972), 11–18

    1972

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    Dixit and A

    A. Dixit and A. Goswami,Combinatorial identities associated with a bivariate generating function for overpartition pairs, Advances in Applied Mathematics 143 (2023), Paper No. 102444

    2023

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    Gasper and M

    G. Gasper and M. Rahman,Basic Hypergeometric Series, second edition, Cambridge Uni- versity Press, Cambridge, 2004

    2004

  7. [7]

    Goswami and R

    A. Goswami and R. Osburn,Quantum modularity of partial theta series with periodic coef- ficients, Forum Mathematicum 33 (2021), no. 2, 451–463. WEIGHTED PARTITIONS AND A MASTER IDENTITY 21 The Pennsylvania State University, University Park, Pennsylvania 16802, USA Email address:andrews@math.psu.edu Department of Mathematical Sciences, United Arab Emirates...

    2021