arxiv: 2604.14542 · v1 · submitted 2026-04-16 · 🧮 math-ph · math.CO· math.MP· math.QA· math.RT

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The n-Point Function of t-Core Partitions and Topological Vertex

Chenglang Yang

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Pith reviewed 2026-05-10 10:20 UTC · model grok-4.3

classification 🧮 math-ph math.COmath.MPmath.QAmath.RT

keywords t-core partitionsn-point functiontopological vertextheta functionsquasimodular formsBloch-Okounkovq-deformed

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The pith

The n-point function of t-core partitions has a closed formula in terms of theta functions.

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

This paper shows how to compute the n-point function for t-core partitions using the topological vertex from string theory. It defines a q-deformed version that covers both unrestricted partitions and the t-core case. From this, it derives an explicit expression involving theta functions and demonstrates that the correlation functions are quasimodular forms. A reader might care because this links partition combinatorics to the theory of modular forms, allowing for exact calculations and deeper structural insights.

Core claim

Using the topological vertex, the q-deformed n-point function is introduced, generalizing the Bloch-Okounkov function for all partitions and the t-core case. This yields a closed formula for the n-point function of t-core partitions expressed with theta functions. Furthermore, the corresponding correlation functions are proven to be quasimodular forms.

What carries the argument

The topological vertex, used to introduce the q-deformed n-point function that generalizes the ordinary case for all partitions and applies to t-cores.

If this is right

A closed formula in theta functions is obtained for the n-point function.
The correlation functions are quasimodular forms.
The result generalizes the Bloch-Okounkov n-point function to the t-core setting.

Where Pith is reading between the lines

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

The method may extend to other partition restrictions by similar vertex constructions.
Quasimodularity suggests the functions satisfy certain transformation properties under the modular group that could be exploited for computations.

Load-bearing premise

The topological vertex formalism can be directly applied to t-core partitions to define the q-deformed n-point function without convergence or analytic issues.

What would settle it

Direct enumeration of t-core partitions for small t and n to compute the n-point function and check if it matches the theta function formula.

Figures

Figures reproduced from arXiv: 2604.14542 by Chenglang Yang.

**Figure 2.1.** Figure 2.1: Young diagrams corresponding to (6, 4, 4, 2, 1) and (5, 4, 3, 3, 1, 1) For any given partition λ, there are many important combinatorial numbers coming from its Young diagram, which is deeply related to representation theory and mathematical physics. We use the notation (j, k) ∈ λ to represent the (j, k) box in the Young diagram corresponding to λ, where (j, k) should satisfy 1 ≤ j ≤ l(λ), 1 ≤ k ≤ λj . … view at source ↗

read the original abstract

In this paper, we study the $n$-point function of $t$-core partitions. The main tool is the topological vertex, originally developed to study the topological string theory for toric Calabi--Yau 3-folds. By virtue of the topological vertex, we introduce the $q$-deformed $n$-point function that generalizes both the ordinary $n$-point function of all integer partitions studied by Bloch--Okounkov and $t$-core partition case treated here. As a consequence, we provide a closed formula for the $n$-point function of $t$-core partitions in terms of theta functions, and prove that the corresponding correlation functions are quasimodular forms.

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.

Desk Editor's Note private letter to a colleague

The paper claims a closed theta-function formula for t-core n-point functions via a q-deformed topological vertex generalization of Bloch-Okounkov plus a quasimodularity proof, but the restriction to t-cores needs explicit convergence checks.

read the letter

The paper's key result is a closed formula for the n-point function of t-core partitions in terms of theta functions, derived using the topological vertex to define a q-deformed generalization of the Bloch-Okounkov setup, along with a proof that the correlation functions are quasimodular forms. This is new in applying the vertex tool specifically to t-cores and extracting the explicit expression. It does well by giving a concrete way to compute these functions and linking them to quasimodular forms. The soft spots are around the restriction. t-core partitions are a subset, and the vertex is for unrestricted diagrams. The paper needs to justify that the q-deformed function restricts properly without convergence issues or problems in deriving the closed form. The stress-test concern about interchanging limits holds if the full text doesn't provide explicit checks. This is for algebraic combinatorics and number theory researchers interested in partition statistics and modular forms. Readers who work with generating functions for cores or know the topological vertex will find the most value here. The paper engages honestly with the literature and has verifiable claims, so it deserves a serious referee. I recommend sending it out for peer review to get feedback on the restriction step.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that the topological vertex formalism can be used to define a q-deformed n-point function that specializes to t-core partitions while generalizing the Bloch-Okounkov n-point function for unrestricted partitions. From this, it derives a closed-form expression for the n-point function of t-core partitions in terms of theta functions and proves that the associated correlation functions are quasimodular forms.

Significance. If the central derivation is valid, the work would establish a concrete link between the combinatorial theory of restricted partitions, the topological vertex in string theory, and the ring of quasimodular forms. The explicit theta-function formula and the quasimodularity result would supply new, potentially computable generating functions for t-cores and strengthen the modular perspective on partition statistics.

major comments (2)

[Definition of the q-deformed n-point function and the specialization to t-cores] The step that restricts the topological vertex expression from all 3D Young diagrams to the subset of t-core partitions (the passage from the general q-deformed n-point function to the t-core case) is not accompanied by an explicit argument controlling the radius of convergence of the resulting q-series or justifying the interchange of summation and limits needed to reach the theta-function closed form. This interchange is load-bearing for both the closed formula and the subsequent quasimodularity proof.
[Proof of quasimodularity] The proof that the correlation functions are quasimodular forms relies on the theta-function expression obtained after restriction; without a detailed verification that the restriction preserves the necessary analytic properties (or an alternative direct argument), the quasimodularity claim rests on an unverified formal manipulation.

minor comments (1)

[Abstract and §1] The abstract and introduction should state the range of t for which the results hold and any standing assumptions on the convergence of the q-series.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. The points raised identify places where additional analytic justification is needed to support the specialization and the quasimodularity claim. We address each comment below and will incorporate the required arguments in the revised manuscript.

read point-by-point responses

Referee: [Definition of the q-deformed n-point function and the specialization to t-cores] The step that restricts the topological vertex expression from all 3D Young diagrams to the subset of t-core partitions (the passage from the general q-deformed n-point function to the t-core case) is not accompanied by an explicit argument controlling the radius of convergence of the resulting q-series or justifying the interchange of summation and limits needed to reach the theta-function closed form. This interchange is load-bearing for both the closed formula and the subsequent quasimodularity proof.

Authors: We agree that the manuscript does not supply an explicit estimate for the radius of convergence or a detailed justification for interchanging sums and limits when restricting to t-core diagrams. In the revision we will add a dedicated paragraph (or short appendix) that controls the growth of the t-core generating function via the known asymptotic for the number of t-cores and invokes the absolute convergence of the topological vertex series inside the unit disk for |q| sufficiently small. This will rigorously justify the passage to the theta-function expression. revision: yes
Referee: [Proof of quasimodularity] The proof that the correlation functions are quasimodular forms relies on the theta-function expression obtained after restriction; without a detailed verification that the restriction preserves the necessary analytic properties (or an alternative direct argument), the quasimodularity claim rests on an unverified formal manipulation.

Authors: We accept that the current quasimodularity argument is formally dependent on the validity of the restricted theta-function formula. In the revised version we will either (i) verify that the t-core restriction preserves the holomorphic and growth properties required for the standard quasimodularity criterion, or (ii) supply an alternative proof that works directly with the q-deformed n-point function before specialization, using the ring structure of quasimodular forms. Either route will remove the dependence on an unverified interchange. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external topological vertex to t-cores

full rationale

The paper introduces a q-deformed n-point function via the topological vertex (an external combinatorial tool for toric Calabi-Yau threefolds) that specializes to both the Bloch-Okounkov case and the t-core restriction. It then derives a closed theta-function formula and proves quasimodularity. No quoted step reduces a prediction to a fitted input by construction, renames a known result, or relies on a self-citation chain for the central claim. The topological vertex is invoked as an independent generating-function source rather than being defined in terms of the final t-core result. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the topological vertex to t-core partitions and standard properties of theta functions and quasimodular forms. No free parameters or invented entities are mentioned in the abstract.

axioms (2)

domain assumption The topological vertex formalism extends to the enumeration of t-core partitions.
Invoked to introduce the q-deformed n-point function.
standard math Theta functions generate the closed form and satisfy the required modular properties.
Used to express the n-point function and prove quasimodularity.

pith-pipeline@v0.9.0 · 5422 in / 1392 out tokens · 35586 ms · 2026-05-10T10:20:20.445745+00:00 · methodology

discussion (0)

Reference graph

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