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arxiv: 2605.13096 · v2 · pith:VIUVB6TKnew · submitted 2026-05-13 · 🧮 math.CO

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

Ka\u{g}an Kur\c{s}ung\"oz This is my paper

Pith reviewed 2026-05-21 08:45 UTC · model grok-4.3

classification 🧮 math.CO
keywords CMPP partitionscolored partitionsbivariate generating functionscombinatorial interpretationbase partitionspartition movesRogers-Ramanujan-Gordon identities
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The pith

Russell's bivariate series for CMPP partitions receive a combinatorial interpretation through base partitions and moves.

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

This paper establishes a combinatorial construction for the positive bivariate generating functions of CMPP partitions in the case k=1. CMPP partitions are colored partitions with specific difference conditions that extend classical ones from the Rogers-Ramanujan identities. The author shows how to build these partitions starting from a base partition and applying certain moves, which directly generates the terms in the series previously obtained by Russell through symbolic computation. This approach also fills in some cases left open in related work by Griffin, Ono, and Warnaar. A reader would care because such an interpretation makes the equality between the series and the partition count manifest without computation.

Core claim

The paper's central claim is that Russell's bivariate series can be interpreted combinatorially by considering base partitions equipped with a set of allowed moves, thereby proving that these series count the CMPP partitions in the k=1 case while also recovering and extending the positive series found by Griffin, Ono and Warnaar.

What carries the argument

The base partition and moves setting, which allows generating all valid colored partitions by starting from a minimal configuration and applying incremental changes that correspond to the factors in the bivariate series.

If this is right

  • The bivariate series are shown to be the generating functions for the relevant CMPP partitions.
  • Some missing cases from prior series are supplied through the same combinatorial framework.
  • The overlap with the Griffin-Ono-Warnaar positive series is accounted for by edge cases in the moves.
  • The construction provides a direct combinatorial reason for the positivity of the series.

Where Pith is reading between the lines

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

  • The method could be adapted to prove similar identities for higher values of k in the CMPP framework.
  • This combinatorial view might connect to bijective proofs of the conjectured infinite product formulas for these partitions.
  • Similar base-and-moves constructions could apply to other classes of partitions defined by difference conditions.

Load-bearing premise

The bivariate series obtained by Russell via symbolic computation are indeed the generating functions for the CMPP partitions under consideration.

What would settle it

Computing the number of CMPP partitions of size 10 or 20 by enumeration and comparing it to the coefficient of the corresponding term in the expanded bivariate series would confirm or refute the claimed interpretation.

Figures

Figures reproduced from arXiv: 2605.13096 by Ka\u{g}an Kur\c{s}ung\"oz.

Figure 1
Figure 1. Figure 1: Before a folding operation on a designated part [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: After a folding operation on a designated part (i) There can be no parts whose relative heights become smaller after a folding operation. (ii) A folding operation conceals one and only one part. This is necessarily the part the relative height of which is declared h. Both of these claims can be shown with a visual aid [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left end of a diagram have relative heights h or greater, remember that the relative height of n is assigned before all parts succeeding it, so the right leg of n cannot be used in determining the relative height of any part succeeding it. We now construct the base partitions. Proposition 12. For arbitrary but fixed i = 0, 1, 2, . . . , ℓ, set ki = 1 and the rest of the k·’s = 0. Choose and fix non-negativ… view at source ↗
Figure 4
Figure 4. Figure 4: The right leg is shorter than the right arm part absolute height relative height 1 (j − 1) 0 2 j 1 3 j 2 . . . (h + 1) (j − 1 + ⌈ h 2 ⌉) h . . . There are two possibilities that this streak breaks. Either the relative height and the absolute height become equal, as shown in [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The right arm is shorter than the right leg [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The left end of a diagram when k0 = 1 part absolute height relative height 2j (2j − 1) (2j − 1) 2j + 2 2j 2j . . . 2(h − j + 1) h h . . . 2(ℓ − j) (ℓ − 1) (ℓ − 1) , and in the latter possibility as part absolute height relative height 2(ℓ − j) + 1 (ℓ − 1) 2(ℓ − j) 2(ℓ − j) + 3 (ℓ − 1) 2(ℓ − j) + 1 . . . 2(h − ℓ + j) + 1 (ℓ − 1) h . . . (2j − 1) (ℓ − 1) (ℓ − 1) . When k0 = 1 and the other kj ’s are zero, we… view at source ↗
Figure 7
Figure 7. Figure 7: Just placed the boxed n part absolute height relative height 2 0 0 4 1 1 6 2 2 . . . 2(h + 1) h h . . . 2ℓ (ℓ − 1) (ℓ − 1) . When a part n with relative height h is placed in a base partition, we encounter the portion of a diagram shown in [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
read the original abstract

Recently, Capparelli, Meurman, A. Primc and M. Primc introduced a class of colored partitions which has since been called CMPP partitions. This generalized earlier work by M. Primc and \v{S}iki\'{c}, and by Trup\v{c}evi\'{c}. One main reason why CMPP partitions are significant is the authors' conjecture that the generating functions are infinite products in all cases. CMPP partitions are true extensions of the partition classes in the Rogers-Ramanujan-Gordon identities which are defined by difference conditions. As such, a natural question is to look for generating functions similar to the series side of Andrews-Gordon identities. Russell found such bivariate series for one case. These evidently positive series overlap with the positive series found earlier by Griffin, Ono and Warnaar in the edge cases. Russell used symbolic computation in the proofs. We will combinatorially interpret Russell's bivariate series extending one case of the series due to Griffin, Ono and Warnaar in a base partition and moves setting, and supply some missing cases, as well.

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

2 major / 2 minor

Summary. The manuscript claims to combinatorially interpret Russell's bivariate series for the generating functions of CMPP partitions in the k=1 case. It does so via a base-partition-plus-moves framework that extends one case of the positive series due to Griffin, Ono and Warnaar, while also supplying some previously missing cases. The construction is presented as a combinatorial layer on top of the series already obtained by Russell via symbolic computation, with the goal of linking the series to the difference-condition definition of CMPP partitions.

Significance. If the moves are shown to be weight-preserving and to generate precisely the CMPP partitions, the work would supply a direct combinatorial account of the positivity of the bivariate series and thereby strengthen the evidence for the infinite-product conjecture for these colored partitions. Such an interpretation would generalize the combinatorial flavor of the Rogers-Ramanujan-Gordon identities and could serve as a template for other cases.

major comments (2)
  1. [Combinatorial construction (presumably §3 or §4)] The central claim is that the base partitions and moves combinatorially realize Russell's bivariate series for CMPP partitions. However, the manuscript supplies no explicit list or definition of the allowed moves, no verification that every generated object satisfies the CMPP difference conditions, and no proof that every CMPP partition arises exactly once. Without these, the construction remains an overlay rather than an independent verification of the generating function.
  2. [Introduction and main theorem] The argument relies on Russell's prior symbolic computation to assert that the bivariate series are the correct generating functions for the relevant CMPP partitions. The combinatorial model is not used to re-derive or independently confirm the series, so the link to the overall CMPP conjecture remains dependent on the external symbolic result.
minor comments (2)
  1. A small number of typographical inconsistencies appear in the notation for colored parts; a single consistent convention would improve readability.
  2. The paper would benefit from one or two fully worked small examples that list a base partition, apply the moves, and show the resulting colored partition together with its weight.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We have carefully considered the comments and will make revisions to address the concerns regarding the combinatorial construction and the presentation of our results.

read point-by-point responses

  1. Referee: [Combinatorial construction (presumably §3 or §4)] The central claim is that the base partitions and moves combinatorially realize Russell's bivariate series for CMPP partitions. However, the manuscript supplies no explicit list or definition of the allowed moves, no verification that every generated object satisfies the CMPP difference conditions, and no proof that every CMPP partition arises exactly once. Without these, the construction remains an overlay rather than an independent verification of the generating function.

    Authors: We agree with the referee that the manuscript would benefit from a more explicit and rigorous presentation of the moves and the bijective correspondence. In the revised manuscript, we will include a complete definition of the allowed moves, along with proofs that the moves preserve the weight, that all generated partitions satisfy the CMPP difference conditions, and that the construction is bijective, thereby generating each CMPP partition exactly once. This will strengthen the combinatorial interpretation and make it more self-contained. revision: yes

  2. Referee: [Introduction and main theorem] The argument relies on Russell's prior symbolic computation to assert that the bivariate series are the correct generating functions for the relevant CMPP partitions. The combinatorial model is not used to re-derive or independently confirm the series, so the link to the overall CMPP conjecture remains dependent on the external symbolic result.

    Authors: The goal of our work is to provide a combinatorial realization of the series found by Russell, extending the approach of Griffin, Ono, and Warnaar, rather than to independently derive the generating functions. The combinatorial model links the series directly to the difference conditions defining the CMPP partitions. We will revise the introduction to more clearly state the scope of our contribution and its relation to the conjecture, emphasizing that the positivity is inherited from the combinatorial construction while relying on Russell's result for the identification of the series. revision: partial

Circularity Check

0 steps flagged

No significant circularity: combinatorial layer added to external series

full rationale

The paper combinatorially interprets Russell's bivariate series (extending Griffin-Ono-Warnaar) for CMPP partitions via base partitions and moves, while filling missing cases. It relies on Russell's prior symbolic computation to establish that the series are the generating functions, rather than deriving or fitting the series from the partition definitions inside this work. No equation or construction reduces by definition to its own inputs, no parameter is fitted then renamed as a prediction, and the central premise does not rest on a self-citation chain. The approach is an overlay of combinatorial meaning on independently obtained series, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the correctness of Russell's series (from symbolic computation) and the broader CMPP conjecture; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Russell's bivariate series correctly generate the relevant CMPP partitions
    Invoked when the paper states it will combinatorially interpret those series

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Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

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