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REVIEW 2 major objections 3 minor 35 references

On a pair of three-colored (mod 10) partition identities

T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves two identities equating the generating functions of restricted three-colored partitions with the products $1/((q;q^2)_\infty(q,q^4;q^5)_\infty)$ and $1/((q;q^2)_\infty(q^2,q^3;q^5)_\infty)$.

desk verdict Solid new identities, but the proof's computational crux is unverifiable from the preprint and one printed atomic relation is misprinted; worth refereeing with revisions. read the letter →

arxiv 2509.07169 v1 pith:ZJU4TCY4 submitted 2025-09-08 math.CO math.NT

Matthew C. Russell This is my paper · ORCID
classification math.COmath.NT MSC 05A1711P8405A15
keywords three-coloredpartitionspartitionidentitiesRogers-Ramanujanq-trinomialcoefficientsatomicrelationsfunctionalequationsgeneratingfunctions
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 establishes a pair of exact product formulas for the generating functions of three-colored partitions into distinct parts, drawn from a set $\Gamma$ with color-dependent difference constraints. One identity forbids the red part 1; the other forbids red 1, red 2, and blue 1. The products are $1/((q;q^2)_\infty(q,q^4;q^5)_\infty)$ and $1/((q;q^2)_\infty(q^2,q^3;q^5)_\infty)$, namely the odd-distinct partition generating function times a Rogers-Ramanujan product. These identities give closed forms for the restricted partition counts, and because the same products arise as principally specialized characters of level-3 standard modules of $A_1^{(1)}$, they tie the colored partitions to affine Lie algebra representation theory. The proof is a template that introduces $q$-trinomial multisums, derives functional equations for bivariate refinements, and uses Maple to check linear atomic-relation combinations.

What carries the argument

The machinery has three parts. First, the $q$-trinomial coefficient $\binom{i+j+k}{i,j,k}_q = (q;q)_{i+j+k}/((q;q)_i(q;q)_j(q;q)_k)$ and the multisums $S_{a,b,c}(x)$ and $T_{a,b,c}(x)$ built from it; these carry the refined counting. Second, the atomic relations rel1–rel9 (and crel1–crel9 for $T$), which are linear relations among shifted multisums proved from $q$-Pascal identities and let the computer rewrite any substituted functional equation as zero. Third, functional equations for the colored-partition generating functions $Q_1,Q_2,Q_3$, untangled by the Murray-Miller algorithm into the single equations (2.15) and (2.16). The paper shows that particular multisum combinations satisfy those same functional equations, then checks initial conditions, forcing equality of generating functions.

What would settle it

Independently rerun the five Maple verifications (or expand both sides of the substituted functional equations by hand or computer and check they reduce to zero); any failure of a reduction would invalidate the proof. Separately, enumerate all three-colored partitions in $\Gamma$ with the stated exclusions up to $n=30$ and compare coefficients with the series expansions of the two product sides; a first mismatch would falsify the corresponding theorem.

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Extended reading notes

Core claim

The central claim is that the two restricted generating functions factor completely: $\sum_{n\ge 0} A(n)q^n = 1/((q;q^2)_\infty(q,q^4;q^5)_\infty)$ (Theorem 1) and $\sum_{n\ge 0} A^*(n)q^n = 1/((q;q^2)_\infty(q^2,q^3;q^5)_\infty)$ (Theorem 2), where $A(n)$ counts three-colored partitions in $\Gamma$ with no $1_R$ and $A^*(n)$ counts those with no $1_R, 2_R, 1_B$. The proof establishes stronger bivariate statements first: the part-count generating functions $Q_1(x)$ and $Q_3(x)$ are identified with explicit combinations of the $T$-multisums, $T_{1,0,1}(x)+xqT_{3,1,2}(x)=Q_1(x)$ and $T_{2,0,1}(x)=Q_3(x)$, and the $S$-multisums are identified with the sum sides of the first and second Rogers-Ramanujan identities via equations (3.1) and (3.5). Setting $x=1$ and applying Euler's odd-distinct identity and the Rogers-Ramanujan identities yields the product forms.

Load-bearing premise

The argument depends on the Maple files MC6 proof A.txt through E.txt confirming that substituted multisum expressions are linear combinations of atomic relations, and those files are not included in the preprint, so the central computational step cannot be checked from the text alone.

Editorial extensions

If this is right

  • The two restricted three-colored partition families have explicit product generating functions, so their coefficients can be computed from the products rather than by enumerating $\Gamma$.
  • The stronger bivariate identities provide information about the number of parts in these partitions, not just their total size.
  • The identities join the Rogers-Ramanujan family, giving colored-partition interpretations of products that also appear as level-3 characters of $A_1^{(1)}$ standard modules.
  • The proof strategy—derive functional equations for bivariate refinements, then use atomic relations to verify the equations—extends to other conjectured Rogers-Ramanujan-type multisum identities, including the refinement proposed in Section 4.

Reading between the lines

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

  • Editorial inference: because the product sides are principally specialized characters of level-3 standard modules of $A_1^{(1)}$, the restricted three-colored partitions may provide a direct combinatorial model for those characters, potentially supporting a crystal-theoretic bijection.
  • Editorial inference: the five cited Maple verifications are essential to the proof, and since the files are not included, an independent computer algebra check of the linear combinations would settle whether the proof's central step is sound.
  • Editorial inference: the method suggests a recipe for proving other conjectured Nahm-sum identities: insert auxiliary variables, derive a bivariate functional equation from a combinatorial interpretation of the product side, and verify the multisum side via atomic relations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. This paper proves two mod-10 partition identities. Theorem 1 states that the generating function for three-colored partitions in Gamma with no part 1_R equals 1/((q;q^2)_infinity (q,q^4;q^5)_infinity), and Theorem 2 states the analogous identity with no parts 1_R, 2_R, 1_B, with product 1/((q;q^2)_infinity (q^2,q^3;q^5)_infinity). The proofs refine both sides by a variable x marking the number of parts, derive functional equations for the refined generating functions Q_1,Q_2,Q_3 of the colored partitions and for two auxiliary pair-of-partitions generating functions, and then use the atomic-relations method to show that the q-trinomial multisums S_{a,b,c} and T_{a,b,c} satisfy the same equations. The algebraic verifications are delegated to Maple files that are referenced but not included; one conjecture refining Theorem 1 is stated in Section 4.

Significance. The identities, if correct, are attractive new examples of colored partition identities with modulus 10, connected to the Rogers-Ramanujan products and to level-3 standard modules of A_1^(1); the atomic-relations method is a promising proof paradigm, and the paper usefully exhibits small cases and prints one full linear combination. The manuscript is honest about its reliance on computer verification and gives enough combinatorial structure that the main functional equations are plausible. However, the current text is not self-contained: a printed atomic relation is false, and the decisive symbolic checks are not reproducible from the submission, so the significance is conditional on a revision that fixes these gaps.

major comments (2)
  1. [Section 2.1] The printed atomic relation crel1 for T_{a,b,c} is false. Its final term is x^2 q^{a+2} T_{a+4,b+2,c+2}, but the shift equation (2.9) and the sentence introducing the T-relations (which says x^2 factors are changed to x) both require x q^{a+2} T_{a+4,b+2,c+2}. For example, with the printed relation, the coefficient of x in crel1_{0,0,0} is q^2 rather than 0. This is load-bearing: the displayed linear combination proving (3.7) in Section 3.2 uses crel1_{2,0,1} and crel1_{2,0,3}, so that combination is incorrect as printed. Please correct the typo and re-verify all identities involving crel1.
  2. [Sections 3.1 and 3.2] The crucial symbolic verifications are not contained in the paper. After display (3.1), (3.3), (3.4), (3.5), and (3.7), the proof asserts that the substituted expression is a linear combination of atomic relations and refers to files 'MC6 proof A.txt' through 'MC6 proof E.txt', but none of these files is included. From the text alone, the central algebraic step is an assertion. Please include the Maple code and output (or an equivalent machine-readable certificate), or display the explicit linear combinations in the paper, so that the proofs are checkable.
minor comments (3)
  1. [Section 2.2] In the justification of (2.12), the bullet for the case where the partition contains 2_R says the rest is counted by Q_1(xq^2), but the displayed term is x q^2 Q_3(xq^2), and the listed forbidden parts support Q_3. Please correct the wording.
  2. [Sections 3.1 and 3.2] The substitutions of T-expressions into (2.15) and (2.16) are said to use (2.8), but the relevant shift equation for T_{a,b,c} is (2.9). Please update these cross-references.
  3. [Section 3.1] After substituting (1+xq)S_{3,0,1}(x) into (3.2), the displayed equation divides by (1+xq); since this factor has constant term 1 as a formal power series, the division is legitimate, but a brief remark would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proofs compare independent generating functions against standard Rogers-Ramanujan products via functional equations and algebraic identities derived from the definitions.

full rationale

The derivation chain is self-contained in the relevant sense. The target product sides enter only at the final step through Euler's odd-distinct identity and the classical Rogers-Ramanujan identities, which are external benchmarks. The bivariate generating functions Q1, Q2, and Q3 are defined combinatorially, and the functional equations (2.12)--(2.14) are derived directly from the forbidden-part conditions of Γ. The Murray-Miller reduction to (2.15) and (2.16) is an algebraic manipulation, not an import of the theorem. The S-type atomic relations are proved in Section 2.1 from the definitions of S_{a,b,c}, and the T-type relations are stated as immediate analogues; they are not adopted as assumed forms of the identities. The key equalities (3.1), (3.3), (3.5), and (3.7) are established by showing that both sides satisfy the same functional equation and agree at initial conditions, with the multisum side verified by explicitly written linear combinations of atomic relations. No parameter is fitted to the claimed product, and no step defines the left-hand object in terms of the desired right-hand product. The cited prior papers on the atomic-relations method are contextual and methodological; the actual relations used here are either proved in the paper or are direct analogues of proved relations. The omission of the Maple verification files and a possible misprint in the printed crel1 relation are real auditability and correctness concerns, but they are not circularity: the proof does not assume the target identities in disguised form.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters; the only assumptions are standard theorems and the established, but not re-proved, atomic-relations and Murray-Miller framework. The decisive Maple computations are an unshipped artifact rather than an axiom.

assumptions (5)
  • standard math Rogers-Ramanujan identities (first and second)
    Used in the final step of Theorems 1 and 2 to convert the distinct-part times Rogers-Ramanujan sum products into their product forms. Cited at [28].
  • standard math Euler's odd-distinct identity
    Identifies (-q;q)_∞ as the generating function for partitions into distinct parts, used in both proofs.
  • domain assumption Atomic relations method yields known linear combination space
    The paper assumes the nine atomic relations are sufficient to express the substituted functional equations. This is the framework from the author's prior work [21, 6], not re-proved in this paper.
  • domain assumption Murray-Miller algorithm untangles the system correctly
    Equations (2.15) and (2.16) are stated to follow from (2.12)-(2.14) via the Murray-Miller algorithm [26], but the derivation is not shown in the text.
  • standard math Uniqueness of solutions of the q-difference functional equations given initial conditions
    Used implicitly to conclude that agreement at x=0 or q=0 plus satisfying the same functional equation implies equality of formal power series.

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Pith. "Pith review of On a pair of three-colored (mod 10) partition identities." pith.science (2026). https://pith.science/paper/ZJU4TCY4

@misc{pith2026250907169,
  author       = {Pith},
  title        = {Pith review of: On a pair of three-colored (mod 10) partition identities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZJU4TCY4}},
  note         = {Machine review of arXiv:2509.07169}
}
read the original abstract

We prove a pair of (mod 10) partition identities. The sum sides involve three-colored partitions into distinct parts, while the product sides are the generating functions for distinct partitions times the Rogers-Ramanujan products. Our proofs make heavy use of Maple to verify that functional equations are satisfied.

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35 extracted references · 33 canonical work pages

  1. [1]

    Refinements and generalizations of Capparelli’s conjecture on partitions.J

    Krishnaswami Alladi, George Andrews, and Basil Gordon. Refinements and generalizations of Capparelli’s conjecture on partitions.J. Algebra, 174(2):636–658, 1995

    work page 1995

  2. [2]

    George E. Andrews. A general theory of identities of the Rogers-Ramanujan type.Bull. Amer. Math. Soc., 80:1033–1052, 1974. 12

    work page 1974

  3. [3]

    George E. Andrews. Schur’s theorem, Capparelli’s conjecture andq-trinomial coefficients. In The Rademacher legacy to mathematics (University Park, PA, 1992), volume 166 ofContemp. Math., pages 141–154. Amer. Math. Soc., Providence, RI, 1994

    work page 1992

  4. [4]

    George E. Andrews. Partition identities for two-color partitions.Hardy-Ramanujan J., 44:74– 80, 2021

    work page 2021

  5. [5]

    Andrews and R

    George E. Andrews and R. J. Baxter. Lattice gas generalization of the hard hexagon model. III.q-trinomial coefficients.J. Statist. Phys., 47(3-4):297–330, 1987

    work page 1987

  6. [6]

    Russell, and Christopher Sadowski

    Katherine Baker, Shashank Kanade, Matthew C. Russell, and Christopher Sadowski. Principal subspaces of basic modules for twisted affine Lie algebras,q-series multisums, and Nandi’s identities.Algebr. Comb., 6(6):1533–1556, 2023

    work page 2023

  7. [7]

    Three-colored partitions and dilated companions of Capparelli’s identities

    Kathrin Bringmann and Karl Mahlburg. Three-colored partitions and dilated companions of Capparelli’s identities. InAnalytic number theory, modular forms andq-hypergeometric series, volume 221 ofSpringer Proc. Math. Stat., pages 139–156. Springer, Cham, 2017

    work page 2017

  8. [8]

    On some double Nahm sums of Zagier

    Zhineng Cao, Hjalmar Rosengren, and Liuquan Wang. On some double Nahm sums of Zagier. J. Combin. Theory Ser. A, 202:Paper No. 105819, 13, 2024

    work page 2024

Show all 35 references
  1. [9]

    Some new modular rank four Nahm sums as lift-dual of rank three examples.arXiv:2508.12468 [math.NT], 2025

    Zhineng Cao and Liuquan Wang. Some new modular rank four Nahm sums as lift-dual of rank three examples.arXiv:2508.12468 [math.NT], 2025

    2025 arXiv

  2. [10]

    On some representations of twisted affine Lie algebras and combinatorial identities.J

    Stefano Capparelli. On some representations of twisted affine Lie algebras and combinatorial identities.J. Algebra, 154(2):335–355, 1993

    1993

  3. [11]

    Linked partition ideals, directed graphs andq-multi-summations.Electron

    Shane Chern. Linked partition ideals, directed graphs andq-multi-summations.Electron. J. Combin., 27(3):Paper No. 3.33, 29, 2020

    2020

  4. [12]

    The Ariki-Koike algebras and Rogers-Ramanujan type partitions.J

    Shane Chern, Zhitai Li, Dennis Stanton, Ting Xue, and Ae Ja Yee. The Ariki-Koike algebras and Rogers-Ramanujan type partitions.J. Algebraic Combin., 60(2):491–540, 2024

    2024

  5. [13]

    Su-Ping Cui and Julia Q. D. Du. Some multi-sum Rogers–Ramanujan type identities with one parameter.Ramanujan J., 68(1):20, 2025

    2025

  6. [14]

    Siladi´ c’s theorem: weighted words, refinement and companion.Proc

    Jehanne Dousse. Siladi´ c’s theorem: weighted words, refinement and companion.Proc. Amer. Math. Soc., 145(5):1997–2009, 2017

    1997

  7. [15]

    Unification, refinements and companions of generalisations of Schur’s theo- rem

    Jehanne Dousse. Unification, refinements and companions of generalisations of Schur’s theo- rem. InAnalytic number theory, modular forms andq-hypergeometric series, volume 221 of Springer Proc. Math. Stat., pages 213–251. Springer, Cham, 2017

    2017

  8. [16]

    On partition identities of Capparelli and Primc.Adv

    Jehanne Dousse. On partition identities of Capparelli and Primc.Adv. Math., 370:107245, 22, 2020

    2020

  9. [17]

    Partition identities from higher level crystals ofA (1) 1 .Proc

    Jehanne Dousse, Leonard Hardiman, and Isaac Konan. Partition identities from higher level crystals ofA (1) 1 .Proc. Amer. Math. Soc., 153(4):1363–1382, 2025

    2025

  10. [18]

    On a Rogers-Ramanujan type identity from crystal base theory.Proc

    Jehanne Dousse and Jeremy Lovejoy. On a Rogers-Ramanujan type identity from crystal base theory.Proc. Amer. Math. Soc., 146(1):55–67, 2018. 13

    2018

  11. [19]

    Shashank Kanade, Debajyoti Nandi, and Matthew C. Russell. A variant of IdentityFinder and some new identities of Rogers-Ramanujan-MacMahon type.Ann. Comb., 23(3-4):807–834, 2019

    2019

  12. [20]

    Russell.IdentityFinderand some new identities of Rogers-Ramanujan type.Exp

    Shashank Kanade and Matthew C. Russell.IdentityFinderand some new identities of Rogers-Ramanujan type.Exp. Math., 24(4):419–423, 2015

    2015

  13. [21]

    Shashank Kanade and Matthew C. Russell. Completing the A 2 Andrews-Schilling-Warnaar identities.Int. Math. Res. Not. IMRN, (20):17100–17155, 2023

    2023

  14. [22]

    Shashank Kanade and Matthew C. Russell. Tight cylindric partitions.arXiv:2508.15113 [math.CO], 2025

    2025 arXiv

  15. [23]

    Rogers-Ramanujan type identities involving double, triple and quadruple sums.Ramanujan J., 65(1):333–390, 2024

    Zhi Li and Liuquan Wang. Rogers-Ramanujan type identities involving double, triple and quadruple sums.Ramanujan J., 65(1):333–390, 2024

    2024

  16. [24]

    Annihilating fields of standard modules ofsl(2,C) ∼ and combinatorial identities.Mem

    Arne Meurman and Mirko Primc. Annihilating fields of standard modules ofsl(2,C) ∼ and combinatorial identities.Mem. Amer. Math. Soc., 137(652):viii+89, 1999

    1999

  17. [25]

    Remarks on Nahm sums for symmetrizable matrices.Ramanujan J., 66(4):Pa- per No

    Yuma Mizuno. Remarks on Nahm sums for symmetrizable matrices.Ramanujan J., 66(4):Pa- per No. 62, 31, 2025

    2025

  18. [26]

    Murray and Kenneth S

    Francis J. Murray and Kenneth S. Miller.Existence theorems for ordinary differential equa- tions. New York University Press, New York, 1954

    1954

  19. [27]

    Some crystal Rogers-Ramanujan type identities.Glas

    Mirko Primc. Some crystal Rogers-Ramanujan type identities.Glas. Mat. Ser. III, 34(54)(1):73–86, 1999

    1999

  20. [28]

    L. J. Rogers. Second Memoir on the Expansion of certain Infinite Products.Proc. Lond. Math. Soc., 25:318–343, 1893/94

  21. [29]

    Matthew C. Russell. A refinement of and a companion to MacMahon’s partition identity. Electron. J. Combin., 32(1):Paper No. 1.4, 10, 2025

    2025

  22. [30]

    Proofs of two conjectural identities on partial Nahm sums

    Changsong Shi and Liuquan Wang. Proofs of two conjectural identities on partial Nahm sums. arXiv:2507.20270 [math.NT], 2025

    2025 arXiv

  23. [31]

    Twistedsl(3,C) ∼-modules and combinatorial identities.Glas

    Ivica Siladi´ c. Twistedsl(3,C) ∼-modules and combinatorial identities.Glas. Mat. Ser. III, 52(72)(1):53–77, 2017

    2017

  24. [32]

    Stembridge

    John R. Stembridge. Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities.Trans. Amer. Math. Soc., 319(2):469–498, 1990

    1990

  25. [33]

    A proof of conjectured partition identities of Nandi

    Motoki Takigiku and Shunsuke Tsuchioka. A proof of conjectured partition identities of Nandi. Amer. J. Math., 146(2):405–433, 2024

    2024

  26. [34]

    Proofs of Mizuno’s conjectures on rank three Nahm sums of index (1,2,2).Adv

    Boxue Wang and Liuquan Wang. Proofs of Mizuno’s conjectures on rank three Nahm sums of index (1,2,2).Adv. Math., 477:Paper No. 110368, 73, 2025

    2025

  27. [35]

    2-colored Rogers-Ramanujan partition identities.Turkish J

    Mohammad Zadehdabbagh. 2-colored Rogers-Ramanujan partition identities.Turkish J. Math., 46(8):3352–3358, 2022. 14

    2022

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