{"id":"b7f950d4-2994-4947-aa40-a32333c694ea","arxiv_id":"2412.14344","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An infinite family of triangular-number recurrences for the 3-colored partition function, plus Euler-type recurrences for 2-colored and all t-regular partition functions, derived by Rankin-Cohen bracket methods.","lead":"This paper writes new Euler-style recurrence formulas for the 2-colored and 3-colored partition functions, and for all t-regular partition functions, expressing each count through smaller counts plus explicit divisor and Hecke-trace corrections. The recurrences extend a recent modular-forms framework of Gomez, Ono, Saad, and Singh from the ordinary partition function to colored partitions, tying partition counting to modular form coefficients.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4's infinite family depends on the unproved hypergeometric reduction in Lemma 2.8 and the deferred Maass-Poincaré identification in Lemma 2.4; until these are verified, the v≥8 recurrences are not established.","rationale":"The elementary parts of the paper—Theorems 1.1, 1.6, and 1.2—are internally consistent and the coefficient comparisons are straightforward. The genuinely load-bearing step is the transition to the infinite family, where the trace formula is the only source of the cuspidal coefficients. The reader identified the same two deferred proofs, and I agree. I sharpen the concern by noting that the v=6 example is not independent confirmation of Proposition 2.5, since β_6 can be obtained by coefficient comparison, and no v≥8 case is checked. The most useful remedy is a symbolic rederivation of Lemma 2.8 plus an end-to-end check at v=8. I am not asserting that the omitted identity is false—only that the infinite-family theorem currently rests on an unverified algebraic step. I therefore keep the reader's CONDITIONAL verdict.","tokens_in":11059,"tokens_out":30644,"duration_ms":249135,"concrete_test":"Independently rederive Lemma 2.8 from Euler's and Pfaff-Saalschütz transformations and verify the claimed identity coefficientwise in 1/n^2 for v=2 and v=8, for example by symbolic computation of both sides and of ω_v(n) in (2.10). If the identity holds, run an end-to-end check of the v=8 case of Theorem 1.4: compute D_f for the unique S_{16} eigenform from (1.9), then evaluate the right-hand side for n=1,...,20 and compare with p3(n) from the generating function 1/(q;q)^3_∞. A mismatch at any of these n shows the recurrence is incorrect; matching at all tested n would substantially mitigate the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the infinite family of triangular recurrences for p3 in Theorem 1.4. For v≥8 the cuspidal correction Tr_{2v}(n) is obtained solely through Proposition 2.5, which identifies ⟨[η^{-3},η^3]_v,f⟩ with ||f||·D_f. The proof of Proposition 2.5 uses two unverified inputs: Lemma 2.4, which identifies 1/η^3 as a weight −3/2 Maass-Poincaré series and defers the proof to [7, Prop. 3.4], and Lemma 2.8, whose second identity is the step that collapses the hypergeometric sums into the ~E_v(j,m) coefficients defining D_f, but whose proof is omitted (\"we leave all the details here\"). If either input is wrong, D_f is miscomputed and Theorem 1.4 fails for every v≥8, while Theorem 1.2, which only uses coefficient comparison in one-dimensional cusp spaces, survives. The numerical check for v=6 does not relieve this: S_{12} is one-dimensional, so β_6 is already fixed by Theorem 1.2, and D_Δ(100,700) merely matches that value rather than independently testing the trace formula. Convergence of the double sums in (1.9) and the summation/integration interchanges in Proposition 2.5 are also asserted rather than proved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives Euler-type recurrences for 2-colored partitions (Theorem 1.1), for all t-regular partitions (Theorem 1.6), and for 3-colored partitions (Theorems 1.2 and 1.4). The method is to apply Rankin--Cohen brackets to 1/η^3 and η^3; comparing Fourier coefficients converts the identity into recurrences involving triangular numbers, divisor functions, and Hecke traces. Theorem 1.2 treats the small weights where the cusp space is zero- or one-dimensional, while Theorem 1.4 states an infinite family valid for weights 2v with v in Z_{≥8} ∪ {6}, with the cuspidal correction expressed as a weighted sum of twisted Dirichlet series over normalized Hecke eigenforms. Theorems 1.1 and 1.6 are proved by elementary q-series coefficient comparison, and the proof of Theorem 1.2 is also direct. The proof of Theorem 1.4 passes through a Maass–Poincaré series identification, a Petersson inner product computation, and a hypergeometric reduction, and the paper currently leaves important parts of that chain unproved or misstated.","tokens_in":11271,"tokens_out":21582,"duration_ms":174682,"significance":"If the missing details are supplied, the paper gives a genuinely new family of exact triangular-number recurrences for p_3(n), parallel to the pentagonal-number framework of Gomez–Ono–Saad–Singh, and it cleanly handles the t-regular case. The elementary recurrences in Theorems 1.1 and 1.6 are immediately usable and their proofs are transparent. The coefficient-comparison strategy in Theorem 1.2 is sound and the stated constants appear internally consistent. However, the full infinite-family claim in Theorem 1.4 is not yet established because it depends on an incorrectly stated multiplier in the Maass–Poincaré identification, an omitted hypergeometric proof, and unaddressed convergence questions. The significance is therefore conditional on those repairs.","major_comments":[{"comment":"The multiplier in the Poincaré series identification appears to be wrong. Under the convention stated in (2.1), η(γz) = ε(γ)(cz+d)^{1/2}η(z), so η^3 has multiplier ε^3 and 1/η^3 has multiplier ε^{-3}. Lemma 2.4 identifies 1/η^3 with P_{[i∞]}(z,8,-3/2,ε^3), and Proposition 2.5 then uses ε^3 for both the δ-factor and η^3. If both factors carry ε^3, the Rankin–Cohen bracket has multiplier ε^6, contradicting Lemma 2.2 and the identity 1/η^3 · η^3 = 1. Concretely, under T: z ↦ z+1, η(z+1)^3 = e^{π i/4}η(z)^3, so 1/η(z+1)^3 = e^{-π i/4} 1/η(z)^3. The Poincaré series and all slash actions in Proposition 2.5 should use ε^{-3} for the 1/η^3 factor. This is load-bearing for the trace formula in Theorem 1.4.","section":"§2, Lemma 2.4 and Proposition 2.5"},{"comment":"The proof of Lemma 2.8 is omitted (“we leave all the details here”). The second identity in this lemma is exactly the step that converts the hypergeometric expression for ω_v(n) into the sums E_v(j,m) defining D_f, so Proposition 2.5 and hence the computation of Tr_{2v}(n) in Theorem 1.4 depend on it. A complete derivation, or a precise reference with the exact statement used, must be supplied; saying that it is “easily adopted” from [7] is not sufficient for a referee to verify the central claim.","section":"§2, Lemma 2.8"},{"comment":"Convergence of the relevant infinite objects is asserted rather than proved. The identity (2.2) represents 1/η^3 as an infinite Poincaré series, and the passage from (2.4) to (2.9) interchanges an infinite sum over γ with an integral over a non-compact domain and then interchanges sums over n and m. In addition, D_f in (1.9) is a double sum over j and m of special values D(f; 2v+2j+2m+2), whose convergence is not justified. Since Theorem 1.4 uses D_f as the cuspidal correction, this missing analysis is load-bearing.","section":"§2, Proposition 2.5 and §1, equation (1.9)"}],"minor_comments":[{"comment":"The v=6 numerical computation is described as a numerical justification of Theorem 1.4, but it is only a consistency check: S_12 is one-dimensional, so β_6 is already determined by the coefficient comparison in Theorem 1.2, and D_Δ(100,700) agreeing with β_6 tests the truncated series, not the trace formula independently.","section":"§1, Example after Theorem 1.4"},{"comment":"The proof of Lemma 2.1 says that “by induction argument” a prime ℓ ≠ 3 exists dividing 8n−1; the induction is not shown. For example, one can use that 8n−1 ≡ 7 (mod 8), so the number cannot be a power of 3. Adding this sentence would make the argument complete.","section":"§2, Lemma 2.1"},{"comment":"There are several typos in this section: “Peterson” should be “Petersson”, “harmonoic” should be “harmonic”, and “the followings are true” should be “the following are true”.","section":"§2, Proposition 2.5"},{"comment":"Reference [7] is an arXiv preprint (arXiv:2411.16968). If the paper relies on Proposition 3.4 and Lemmas 3.12/3.13 of [7] as heavily as it does, the authors should state explicitly which results are quoted from the preprint and verify that the preprint version is stable and publicly available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The elementary parts of the paper (Theorems 1.1, 1.6, and 1.2) are likely correct and suitable for publication once the presentation is cleaned up. The main uncertainty is Theorem 1.4, which depends on the Maass–Poincaré identification with the correct multiplier, the omitted hypergeometric details, and convergence justifications. These are fixable within the scope of a revision, so I recommend major revision rather than rejection, but I would want to see a revised version before endorsing the infinite-family claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: this is a competent and honest extension of the Gomez–Ono–Saad–Singh pentagonal-number framework to 2- and 3-colored partitions and t-regular partitions. The genuinely new content is the infinite family of triangular-number recurrences for p3 in Theorem 1.4, with explicit sigma_{2v-1} and Hecke-trace corrections, plus the explicit small-v recurrences in Theorem 1.2. The elementary recurrences in Theorems 1.1 and 1.6 are immediate from Jacobi's identity and Euler's pentagonal theorem, and they are not the selling point; the selling point is the trace formula.\n\nWhat the paper does well: the derivations are transparent, the constants alpha_v and beta_v are obtained by coefficient comparison rather than fitted to data, and I spot-checked the p2 recurrence and the v=0 case; everything is internally consistent. Lemma 2.1's nonvanishing argument is cute and correct. The paper is also honest about what is borrowed from [7].\n\nThe soft spots are real but concentrated. Theorem 1.4 for v>=8 depends on Proposition 2.5, which in turn rests on two unproved inputs: Lemma 2.4 identifies 1/eta^3 as a Maass–Poincaré series with proof deferred to [7, Prop. 3.4], and Lemma 2.8, the Pfaff–Saalschütz hypergeometric reduction, is stated with 'we leave all the details here.' That second lemma is load-bearing: if it is wrong, D_f is miscomputed and Theorem 1.4 fails for every v>=8, while Theorem 1.2 survives because it only uses coefficient comparison in one-dimensional cusp spaces. The numerical check for v=6 does not relieve this: S_12 is one-dimensional, so beta_6 is already fixed by Theorem 1.2, and the truncated sum merely matches that value rather than independently testing the trace formula. Convergence of the double sums in (1.9) and the summation/integration interchanges are also asserted rather than proved.\n\nNone of this is fatal. The missing proofs are likely routine adaptations of [7], and the structure strongly suggests the theorem is true. But as written, the infinite family is conditional on unstated computations.\n\nWho this is for: partition theorists and people working on Rankin–Cohen brackets and harmonic Maass forms. A serious referee can handle it. I would send it out, with a request that the authors supply the details of Lemmas 2.4 and 2.8 (or precise statements with page references) and add at least one v>=8 numerical check.","headline":"A competent extension of the Gomez–Ono–Saad–Singh framework to t-colored and t-regular partitions, whose genuinely new infinite family of p3 recurrences is conditional on two unproved lemmas; the elementary recurrences are solid and correctly derived.","tokens_in":11962,"tokens_out":1943,"would_cite":true,"duration_ms":15938,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A17","11F11","11F25","11P81"],"pacs":[],"model":"deepseek-v4-flash","headline":"The 3-colored partition function obeys an infinite family of exact triangular-number recurrences, with corrections built from divisor sums and Hecke traces.","keywords":["Euler-type recurrence","3-colored partition","triangular number","Rankin–Cohen bracket","Hecke trace","t-regular partition","modular form","partition function"],"falsifier":"Compute the right-hand side of Theorem 1.4 for $v=8$ and $n=5$, using the stated definition of $\\operatorname{Tr}_{16}(5)$ with truncated sums over $m$ and over eigenforms, and compare it with $p_3(5)=108$; any nonzero discrepancy would falsify the general recurrence. As a lighter check, recompute $D_\\Delta$ for the weight-12 case with the paper's own summation ranges and compare with $\\beta_6=-51051/22112$, which tests the convergence assumptions of Proposition 2.5 without invoking higher-weight cusp data.","tokens_in":10732,"feed_emoji":"🔺","tokens_out":13720,"duration_ms":106534,"temperature":0.7,"pith_summary":"This paper establishes Euler-type recurrences for t-colored partitions (t = 2 and 3) and for t-regular partitions of every order. Its central claim is an infinite family of 'triangular number' recurrences for the 3-colored partition function p3(n): for each weight v in {6} ∪ Z_{≥8}, the value p3(n) is expressed in terms of earlier values p3(n − T_k) at triangular numbers T_k, corrected by a divisor-sum term and by the Hecke trace Tr_{2v}(n). The recurrences are proved by forming the v-th Rankin–Cohen bracket of 1/η(z)^3 with η(z)^3, showing it is a holomorphic modular form of weight 2v and level 1, and then comparing Fourier coefficients. Explicit small-weight versions of the recurrences are worked out, with correction terms reducing to divisor sums and to Ramanujan-type coefficients, together with a pentagonal-number recurrence for 2-colored partitions and for all t-regular partitions.","feed_headline":"Triangular-number recurrences found for 3-colored partitions","feed_subtitle":"New recurrences express p3(n) through earlier values, divisor sums, and Hecke traces.","key_machinery":"The central object is the $v$-th Rankin–Cohen bracket $R_v(z):=[1/\\eta(z)^3,\\eta(z)^3]_v$, a bilinear combination of the weight $-3/2$ form $1/\\eta^3$ and the weight $3/2$ form $\\eta^3$ and their derivatives. Lemma 2.2 shows $R_v$ is a holomorphic modular form of weight $2v$ and level $1$, with $q$-expansion $$R_v(z)=\\sum_{n\\ge 0}\\left(\\sum_{k\\ge 0}(-1)^k E_v(n,k)\\,p_3(n-T_k)\\right)q^n.$$ Since the space of weight-$2v$ level-$1$ modular forms is spanned by the Eisenstein series $E_{2v}$ and the cusp forms $S_{2v}(1)$, comparing coefficients of $q^n$ produces the recurrences once the cuspidal part is identified. The load-bearing identification is Proposition 2.5, which evaluates the Petersson inner product $\\langle R_v,f\\rangle$ as $\\|f\\|\\cdot D_f$, where $D_f$ is a weighted infinite sum of special values of twisted Dirichlet series associated to $f$; the proof passes through a Maass–Poincaré series representation of $1/\\eta^3$, an integral evaluation using M-Whittaker functions, and Euler and Pfaaff–Saalschütz hypergeometric reductions.","core_discovery":"On the paper's own terms, the discovery is that the generating function identity $$1/(q;q)_\\$infty^{3}$ = $q^{{1/8}}$/\\eta(z)^3$$ combines with the Jacobi triple product expansion $\\eta(z)^3 = q^{1/8}\\sum_{k\\ge 0}(-1)^k(2k+1)q^{T_k}$ to produce, for every $v\\in\\{6\\}\\cup\\mathbb{Z}_{\\ge 8}$ and every positive integer $n$, the exact recurrence of Theorem 1.4: $$p_3(n)=\\frac{1}{E_v(n,0)}\\left(-\\frac{4v\\,E_v(0,0)}{B_{2v}}\\,\\sigma_{2v-1}(n)+\\operatorname{Tr}_{2v}(n)+\\sum_{k\\ge 1}(-1)^{k+1}E_v(n,k)\\,p_3(n-T_k)\\right).$$ Here $E_v(n,k)$ are the explicit rational coefficients coming from the Rankin–Cohen bracket expansion, $B_{2v}$ is a Bernoulli number, $\\sigma_{m}$ is a divisor sum, and $\\operatorname{Tr}_{2v}(n)$ is a weighted sum of Fourier coefficients of normalized Hecke eigenforms of weight $2v$. The paper also proves the explicit small-weight recurrences of Theorem 1.2, the 2-colored recurrence of Theorem 1.1, and the $t$-regular recurrences of Theorem 1.6. The core claim is that comparing the modular-form expansion of the bracket against Eisenstein series and cusp forms yields these recurrences uniformly in $v$, with the cuspidal part exactly a Hecke trace.","pith_inferences":["If the general recurrence holds, the Hecke trace $\\operatorname{Tr}_{2v}(n)$ is an effectively computable arithmetic quantity; the paper's $v=6$ numerical check ($D_\\Delta(100,700)=-2.308746\\ldots$) suggests the defining double sums converge quickly, so truncation bounds for $D_f$ would turn the recurrences into practical algorithms for large $n$.","The modular interpretation suggests that $p_3(n)$ may inherit congruence properties from the Hecke eigenvalues, parallel to Ramanujan-type congruences for $p(n)$; reducing the $v=2$ recurrence modulo small primes is a direct way to test this.","The Rankin–Cohen bracket construction may extend to other eta quotients $1/\\eta^t$ whose reciprocal is a harmonic Maass form of negative weight, yielding analogous recurrences for other colored partition functions beyond $t=2,3$.","The main unproved-in-detail steps (the Poincaré-series representation of $1/\\eta^3$ and the hypergeometric reduction in Lemma 2.8) are exactly the places where an independent derivation or an explicit error bound would be needed to make the infinite family fully self-contained."],"forward_implications":["For $v\\in\\{2,3,4,5,7\\}$, Theorem 1.2 gives explicit recurrences for $p_3(n)$ whose only arithmetic input beyond earlier partition values is the divisor sum $\\sigma_{2v-1}(n)$.","For $v\\in\\{6,8,9,10,11,13\\}$, the recurrences gain a correction term $\\beta_v\\,\\tau_{2v}(n)$ from the unique normalized cusp form of weight $2v$.","Theorem 1.4 extends the family to every $v\\ge 8$, with the cuspidal correction given by the Hecke trace $\\operatorname{Tr}_{2v}(n)$, so the recurrences exist for all even weights at least 16 (and weight 12).","Theorem 1.6 gives every $t$-regular partition function a pentagonal-number recurrence, with a correction term only when $n=t\\,w_j$ for some pentagonal number $w_j$.","Lemma 2.1 guarantees $E_v(n,0)\\neq 0$ for all $n\\ge 1$, so each recurrence genuinely solves for $p_3(n)$."],"supporting_citations":[{"why":"Supplies the framework for deriving recurrences from Rankin–Cohen brackets and the deferred proof of the Poincaré-series representation of 1/η^3.","marker":"[7]"},{"why":"Provides the Rankin–Cohen bracket formalism and modularity properties used in Lemma 2.2 and Proposition 2.3.","marker":"[3]"},{"why":"The original source for Rankin–Cohen brackets and differential operators on modular forms.","marker":"[12]"},{"why":"Gives the lemma used to show the Poincaré series is a harmonic Maass form, a step in Proposition 2.5.","marker":"[4]"},{"why":"Supplies Euler's and Pfaaff–Saalschütz hypergeometric identities used in Lemma 2.8.","marker":"[2]"},{"why":"Used for the M-Whittaker function derivative and integral identities in Lemmas 2.6 and 2.7.","marker":"[10]"},{"why":"Provides the q-expansion identity η(z)^3 = Σ (-4/n) n q^{n^2/8} used in Proposition 2.5.","marker":"[9]"},{"why":"Background for Euler's pentagonal number theorem and the classical partition recurrences that are being generalized.","marker":"[1]"}],"fun_headline_variants":["Infinite triangular-number recurrences for 3-colored partitions","Hecke traces and divisor sums give exact partition recurrences","New recurrences for 3-colored and all t-regular partitions","Triangular-number recurrences extend to all t-regular partitions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything beyond the small-weight recurrences rests on Proposition 2.5, which identifies the Petersson inner product with a weighted Dirichlet-series sum; that identification assumes the Poincaré-series representation of $1/\\eta^3$ and the hypergeometric reduction are correct (both are quoted rather than proved here), and assumes the double sums defining $D_f$ converge.","fun_headline_variants_meta":{"raw":{"variants":["Infinite triangular-number recurrences for 3-colored partitions","Hecke traces and divisor sums give exact partition recurrences","New recurrences for 3-colored and all t-regular partitions","Triangular-number recurrences extend to all t-regular partitions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001923,"raw_usage":{"total_tokens":7553,"prompt_tokens":993,"completion_tokens":6560,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":609,"completion_tokens_details":{"reasoning_tokens":6490}},"tokens_in":609,"tokens_out":6560,"duration_ms":40281,"temperature":1.0,"reasoning_tokens":6490,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:20:52.308190+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the right-hand side of Theorem 1.4 for $v=8$ and $n=5$, using the stated definition of $\\operatorname{Tr}_{16}(5)$ with truncated sums over $m$ and over eigenforms, and compare it with $p_3(5)=108$; any nonzero discrepancy would falsify the general recurrence. As a lighter check, recompute $D_\\Delta$ for the weight-12 case with the paper's own summation ranges and compare with $\\beta_6=-51051/22112$, which tests the convergence assumptions of Proposition 2.5 without invoking higher-weight cusp data.","supporting_citations":[{"cited_title":"Pentagonal number recurrence relations for $p(n)$","cited_arxiv_id":"2411.16968","evidence_quote":"Supplies the framework for deriving recurrences from Rankin–Cohen brackets and the deferred proof of the Poincaré-series representation of 1/η^3."},{"cited_title":"Bringmann, A","cited_arxiv_id":null,"evidence_quote":"Provides the Rankin–Cohen bracket formalism and modularity properties used in Lemma 2.2 and Proposition 2.3."},{"cited_title":"Zagier, Modular forms and differential operators , Proc","cited_arxiv_id":null,"evidence_quote":"The original source for Rankin–Cohen brackets and differential operators on modular forms."},{"cited_title":"Bringmann and K","cited_arxiv_id":null,"evidence_quote":"Gives the lemma used to show the Poincaré series is a harmonic Maass form, a step in Proposition 2.5."},{"cited_title":"Andrews, R","cited_arxiv_id":null,"evidence_quote":"Supplies Euler's and Pfaaff–Saalschütz hypergeometric identities used in Lemma 2.8."},{"cited_title":"http://dlmf.nis t.gov/, Release 1.0.19 of 2018-06-22","cited_arxiv_id":null,"evidence_quote":"Used for the M-Whittaker function derivative and integral identities in Lemmas 2.6 and 2.7."},{"cited_title":"Ono, The web of modularity: Arithmetic of the coefficients of modul ar forms and q-series, CBMS, Regional Conference series in Mathematics, 102, Amer","cited_arxiv_id":null,"evidence_quote":"Provides the q-expansion identity η(z)^3 = Σ (-4/n) n q^{n^2/8} used in Proposition 2.5."},{"cited_title":"Andrews, The theory of partitions , Cambridge University Press, Cambridge, 1984","cited_arxiv_id":null,"evidence_quote":"Background for Euler's pentagonal number theorem and the classical partition recurrences that are being generalized."}],"review_version":1}