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Congruences for an analogue of Lin's partition function

T0 review · 0 major / 2 minor · reviewed 2026-05-18 · grok-4.3

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Pith's one-line read An analogue B(n) of Lin's restricted partition function satisfies Ramanujan-type congruences for certain sums modulo 2, 3, 5, 7, and 9.

desk verdict This paper defines a new restricted partition function B(n) as an analogue of Lin's and proves congruences for certain sums of it modulo 2, 3, 5, 7, and 9 via standard q-series methods. read the letter →

arxiv 2510.13685 v2 submitted 2025-10-15 math.NT math.CO

Russelle Guadalupe This is my paper · ORCID
classification math.NTmath.CO
keywords partitionfunctionRamanujancongruencesq-seriesmodularfunctionsrestrictedpartitionsgenerating
checked against Foundation.ArithmeticFromLogic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper defines B(n) to count partition triples (π1, π2, π3) of n where π1 and π2 have distinct odd parts and π3 consists of parts divisible by 4. It then proves that selected sums involving these values obey congruences modulo 2, 3, 5, 7, and 9. The proofs rely on elementary q-series manipulations and properties of modular functions applied to the generating function of B(n). A reader would care because the results supply fresh examples of arithmetic regularity in a new family of restricted partitions, extending the classical Ramanujan congruences in a concrete and checkable way.

What carries the argument

The generating function of B(n), which is shown to admit a form permitting direct application of q-series identities and modular-function properties to produce the claimed congruences.

What would settle it

Compute the relevant sum of B(n) for a concrete arithmetic progression and modulus (for example n ≡ 1 mod 5) and check whether it violates the stated congruence for any n up to a few hundred.

Watch

Extended reading notes

Core claim

We introduce B(n) counting the number of partition triples (π1, π2, π3) of n such that π1 and π2 comprise distinct odd parts and π3 consists of parts divisible by 4. Using elementary q-series techniques and modular functions we establish Ramanujan-type congruences modulo 2, 3, 5, 7, and 9 for certain sums involving B(n).

Load-bearing premise

The generating function for B(n) can be written in a form that permits direct application of q-series identities and modular-function properties.

Editorial extensions

If this is right

  • Linear combinations of B(n) vanish or satisfy simple divisibility conditions in specified residue classes modulo 2, 3, 5, 7, or 9.
  • The generating function of B(n) obeys modular relations that mirror those satisfied by classical partition generating functions.
  • The same q-series approach yields explicit recurrence or closed-form expressions for the sums that appear in the congruences.
  • B(n) provides a new concrete family of restricted partitions whose values are constrained by modular arithmetic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the generating function of B(n) can be expressed as an eta-product or similar modular form, the same method may produce congruences for additional moduli or for related counting functions.
  • The existence of these congruences suggests that density or distribution results for B(n) in arithmetic progressions can be obtained by standard circle-method or modular-form techniques.
  • The construction may be varied by altering the divisibility condition on π3 or the odd-part restrictions on π1 and π2, potentially producing further families with analogous arithmetic properties.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript defines B(n) as the number of partition triples (π1, π2, π3) of n where π1 and π2 consist of distinct odd parts and π3 consists of parts divisible by 4. It then uses elementary q-series techniques and properties of modular functions to establish Ramanujan-type congruences modulo 2, 3, 5, 7, and 9 for certain (unspecified in the abstract) sums involving B(n).

Significance. If the derivations hold, the work extends the study of arithmetic properties of restricted partition functions by producing explicit new congruences for this analogue of Lin's function. The reliance on standard q-series and modular-form methods is appropriate for the field and yields falsifiable predictions that can be checked computationally for small n.

minor comments (2)
  1. The abstract and introduction refer to 'certain sums involving B(n)' without an explicit definition or notation for these sums (e.g., S_k(n) or similar); this should be introduced with a clear formula early in the paper.
  2. A short table of small values of B(n) and the relevant sums would aid readability and allow immediate verification of the claimed congruences for small moduli.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment. The summary accurately reflects the definition of B(n) and the methods employed. We are pleased that the referee views the congruences as extending prior work on restricted partition functions and that the results are computationally verifiable. Since the recommendation is for minor revision and no specific major comments were raised, we will incorporate clarifications to improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained via external identities

full rationale

The paper defines the function B(n) combinatorially as the number of partition triples with specified restrictions on parts, yielding a generating function that is a product of standard q-series for distinct odd parts and 4-divisible parts. Congruences are then derived using elementary q-series techniques and properties of modular functions, which are invoked as established external tools rather than derived internally or fitted to the target results. No self-citations are load-bearing for the central claims, no parameters are fitted and renamed as predictions, and no step reduces by construction to its own inputs. The argument chain remains independent of the specific congruences being proved.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the definition of B(n) together with the domain assumption that its generating function is amenable to standard q-series and modular-function manipulations; no free parameters, invented entities, or ad-hoc axioms are indicated.

assumptions (1)
  • domain assumption The generating function for B(n) admits a representation that allows application of q-series identities and modular form properties to derive the congruences.
    Invoked throughout the proof strategy described in the abstract.

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Cite this review

Pith. "Pith review of Congruences for an analogue of Lin's partition function." pith.science (2026). https://pith.science/paper/2510.13685

@misc{pith2026251013685,
  author       = {Pith},
  title        = {Pith review of: Congruences for an analogue of Lin's partition function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2510.13685}},
  note         = {Machine review of arXiv:2510.13685}
}
abstract

We study certain arithmetic properties of an analogue $B(n)$ of Lin's restricted partition function that counts the number of partition triples $\pi=(\pi_1,\pi_2,\pi_3)$ of $n$ such that $\pi_1$ and $\pi_2$ comprise distinct odd parts and $\pi_3$ consists of parts divisible by $4$. With the help of elementary $q$-series techniques and modular functions, we establish Ramanujan-type congruences modulo $2,3,5,7$, and $9$ for certain sums involving $B(n)$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Partitions with parity restrictions: a bijective approach

    math.CO 2026-07 accept novelty 6.0 of 10

    Many generating-function identities for parity-restricted partitions (and related overpartitions, mock theta coefficients, and tableaux) admit direct bijective proofs, some simpler than the original algebraic ones.

Reference graph

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