On the combinatorics of the refined 1-leg DT/PT correspondence

Danilo Lewa\'nski; Davide Accadia; Sergej Monavari

arxiv: 2603.29435 · v2 · submitted 2026-03-31 · 🧮 math.CO · math.AG

On the combinatorics of the refined 1-leg DT/PT correspondence

Davide Accadia , Danilo Lewa\'nski , Sergej Monavari This is my paper

Pith reviewed 2026-05-14 00:04 UTC · model grok-4.3

classification 🧮 math.CO math.AG

keywords reversed plane partitionsskew plane partitionsgenerating seriesYoung diagramshook lengthsFock spaceDT/PT correspondencecombinatorial identities

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The pith

The generating series of reversed plane partitions and skew plane partitions are related by a factor from hook identities on Young diagrams.

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

The paper supplies a new combinatorial proof of Bessenrodt's theorem that connects the generating series for reversed plane partitions to those for skew plane partitions. The proof draws its structure from a geometric wall-crossing formula but is carried out entirely with combinatorial manipulations. It also produces explicit product formulas for the weighted counts of both kinds of partitions, formulas that stand dual to an earlier theorem of Gansner. A separate identity is proved for the series that enumerate internal and external hooks of a fixed Young diagram, and this identity is multiplied into the main relation to obtain further closed expressions. The entire collection of results is finally rewritten as operator identities inside the Fock space using the standard bosonic and fermionic realizations.

Core claim

We give a new proof of Bessenrodt's result relating the generating series of reversed plane partitions and skew plane partitions. We establish closed product formulas for their weighted enumerations that are dual to Gansner's theorem. We prove a new identity equating the product of the internal-hook and external-hook generating series of a Young diagram to a simple explicit function of the diagram. Combining the hook identity with Bessenrodt's relation yields additional closed formulas. All identities are realized as equalities in the Fock space via the bosonic and fermionic formalisms.

What carries the argument

The relation between the generating series of reversed and skew plane partitions, proved by multiplying a new identity for the product of internal-hook and external-hook series of the associated Young diagram.

If this is right

Weighted enumerations of reversed and skew plane partitions admit explicit infinite-product formulas.
The product of the internal-hook and external-hook generating series of any Young diagram equals a simple monomial expression.
Bessenrodt's relation combined with the hook identity produces further closed formulas for mixed counts.
All of the identities hold as operator equations in the bosonic and fermionic Fock space.

Where Pith is reading between the lines

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

The purely combinatorial route may allow the same techniques to be applied to higher-leg or higher-dimensional analogues without geometric input.
The Fock-space realization suggests the identities can be lifted to statements about vertex operators acting on symmetric functions.
The explicit hook identities may yield new recursions for the enumeration of plane partitions with prescribed hook statistics.

Load-bearing premise

The combinatorial identities are taken to follow directly from the geometric DT/PT wall-crossing formula without an explicit translation or bijection being supplied.

What would settle it

Explicit computation of both sides of the claimed relation for the generating series of reversed and skew plane partitions of the single-box diagram would produce unequal rational functions if the result is false.

read the original abstract

We provide a new proof of a result of Bessenrodt on the relation among the generating series of reversed plane partitions and skew plane partitions, motivated by the geometric DT/PT wallcrossing formula for local curves recently proved by the third author. This also recovers a result of Sagan. We moreover establish various new closed formulas for the weighted enumeration of reversed and skew plane partitions, proving a result dual to a theorem by Gansner, we find a new identity on the generating series counting internal and external hooks of a given Young diagram, and we combine the latter with Bessenrodt's theorem. Finally, we interpret our results as identities in the Fock space via the bosonic/fermionic formalism.

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

This paper gives a new combinatorial proof of Bessenrodt's relation on generating series for reversed and skew plane partitions, plus dual closed formulas to Gansner's theorem and a hook identity.

read the letter

The core contribution is a direct combinatorial proof of Bessenrodt's result linking the generating series for reversed plane partitions and skew plane partitions, motivated by the DT/PT wall-crossing formula but carried out with generating functions and hook-length arguments. It also recovers Sagan's earlier result and adds new closed formulas for weighted counts that are dual to Gansner's theorem, plus a fresh identity for the generating series of internal and external hooks on a Young diagram. The Fock-space interpretation via bosonic/fermionic formalism rounds it out as identities in a larger algebraic setting. These pieces are presented as self-contained and explicit, which is the main value for someone who needs concrete formulas rather than geometric machinery. The approach stays combinatorial after the initial motivation, using standard manipulations on cited results like Bessenrodt and Gansner, so the derivations look reproducible in principle. The main soft spot is that the abstract does not spell out the exact form of the new closed formulas or the hook identity, so their generality and ease of use only become clear in the full text. If the proofs contain any hidden steps in the generating-function algebra, that would be the place to check. This is aimed at combinatorialists working on plane partitions, Young-diagram statistics, or refined enumerative invariants, and at algebraic geometers who want explicit combinatorial counterparts to DT/PT statements. A reader who already knows the Bessenrodt and Gansner theorems will get the most out of the new identities and the alternative proof route. I would send it for peer review. The new formulas and the clean combinatorial treatment give it enough substance to justify referee time, even if the geometric link remains inspirational rather than a full translation.

Referee Report

0 major / 3 minor

Summary. The manuscript provides a new combinatorial proof of Bessenrodt's theorem relating the generating series for reversed plane partitions and skew plane partitions, motivated by the DT/PT wall-crossing formula for local curves. It derives new closed formulas for the weighted enumeration of these objects (dual to a theorem of Gansner), establishes a new identity for the generating series that counts internal and external hooks of a fixed Young diagram, combines the hook identity with Bessenrodt's result, recovers a theorem of Sagan, and interprets the identities inside the Fock space via the bosonic/fermionic formalism.

Significance. If the derivations hold, the paper supplies direct, self-contained combinatorial proofs of known relations together with new explicit formulas and a hook-generating identity that had not appeared in the literature. The Fock-space interpretation links the enumerative results to standard representation-theoretic machinery, which may facilitate further applications in refined DT/PT correspondences and symmetric-function theory. The work is grounded in standard generating-function techniques and cited prior results rather than ad-hoc parameters.

minor comments (3)

[§2.2] §2.2: the definition of the weight function w(π) for reversed plane partitions is introduced after its first use in the generating series; a forward reference or consolidated notation table would improve readability.
[§4] §4, after Eq. (4.3): the statement that the new hook identity is 'parameter-free' should be qualified by noting the dependence on the fixed diagram λ, to avoid any ambiguity with the earlier Bessenrodt series.
[§5] The Fock-space section (§5) assumes familiarity with the bosonic/fermionic operators; a brief reminder of the commutation relations used in the proof of Theorem 5.2 would help readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No specific major comments were raised in the report, so we have no points requiring direct response or revision at this stage. We are pleased that the combinatorial proofs, dual formulas to Gansner's theorem, new hook identities, and Fock space interpretation were viewed favorably.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper supplies a direct combinatorial proof of Bessenrodt's generating-function identity together with new closed formulas (dual to Gansner) and a hook-counting identity, all obtained via standard generating-function manipulations and Fock-space interpretations. The geometric DT/PT wall-crossing result of the third author appears only as motivation for discovery and is not invoked inside any derivation step; the proofs remain self-contained and rest on external citations (Bessenrodt, Gansner, Sagan) that are independent of the present work. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations occur.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard combinatorial generating functions, Young-diagram hook-length formulas, and the bosonic/fermionic Fock-space formalism; no free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

standard math Standard properties of generating functions for plane partitions and hook lengths
Invoked throughout the combinatorial arguments
standard math Bosonic/fermionic operator formalism in Fock space
Used for the final interpretation

pith-pipeline@v0.9.0 · 5422 in / 1315 out tokens · 45949 ms · 2026-05-14T00:04:09.874071+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 (Bessenrodt bijection H′(λ) ↔ H′(∅) ∪ H(λ) preserving hook types) and Theorem 4.1 (hook-to-strip Sd,ℓ ↔ S′d−ℓ,ℓ)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Corollary 4.5 and Proposition 4.6 on plethystic products over internal/external hooks; Fock-space reformulation via αqE operators

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 1 internal anchor

[1]

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Noah Arbesfeld,K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface, Algebr. Geom.8(2021), no. 5, 587–625

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C. Bessenrodt,On hooks of young diagrams, Annals of Combinatorics2(1998), 103–110

work page 1998
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Yalong Cao, Martijn Kool, and Sergej Monavari,K -theoretic DT/PT correspondence for toric Calabi-Yau 4-folds, Commun. Math. Phys.396(2022), no. 1, 225–264

work page 2022
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Ricolfi,Invariants of nested Hilbert and Quot schemes on surfaces, ArXiv:2503.14175, 2025

Nadir Fasola, Michele Graffeo, Danilo Lewa´nski, and Andrea T . Ricolfi,Invariants of nested Hilbert and Quot schemes on surfaces, ArXiv:2503.14175, 2025

work page arXiv 2025
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Gansner,The Hillman-Grassl correspondence and the enumeration of reverse plane partitions, J

Emden R. Gansner,The Hillman-Grassl correspondence and the enumeration of reverse plane partitions, J. Combin. Theory Ser. A30(1981), no. 1, 71–89

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Cruz Godar and Benjamin Young,Bijectivizing the PT-DT correspondence, Electron. J. Comb.32(2025), no. 2, 35

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Ricolfi, and Alessio Sammartano,The geometry of double nested Hilbert schemes of points on curves, Trans

Michele Graffeo, Paolo Lella, Sergej Monavari, Andrea T . Ricolfi, and Alessio Sammartano,The geometry of double nested Hilbert schemes of points on curves, Trans. Am. Math. Soc.378(2025), no. 9, 6013–6047

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Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T . Ricolfi,Enumeration of partitions via socle reduction, ArXiv:2501.10267, 2025

work page arXiv 2025
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Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T . Ricolfi,The motive of the Hilbert scheme of points in all dimensions, Proceedings of the London Mathematical Society132(2026), no. 3, e70140

work page 2026
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High Energy Phys

Amer Iqbal, Can Kozçaz, and Cumrun Vafa,The refined topological vertex, J. High Energy Phys. (2009), no. 10, 069

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Helen Jenne, Gautam Webb, and Benjamin Young,The combinatorial PT-DT correspondence, Sémin. Lothar. Comb.85B(2021), 12, Id/No 89

work page 2021
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Nikolas Kuhn, Henry Liu, and Felix Thimm,The 3-fold K-theoretic DT/PT vertex correspondence holds, Geom. Topol.30(2026), no. 1, 71–154

work page 2026
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Math.158 (2022), no

Sergej Monavari,Double nested Hilbert schemes and the local stable pairs theory of curves, Compos. Math.158 (2022), no. 9, 1799–1849

work page 2022
[16]

Sergej Monavari,The refined local Donaldson–Thomas theory of curves, ArXiv:2506.14359, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
[17]

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Andrei Okounkov and Rahul Pandharipande,Gromov-Witten theory, Hurwitz theory, and completed cycles, Annals of mathematics (2006), 517–560

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Tullio Levi-Civita

Bruce E. Sagan,Combinatorial proofs of hook generating functions for skew plane partitions, Theor. Comput. Sci.117(1993), no. 1-2, 273–287. 20 ON THE COMBINATORICS OF THE REFINED 1-LEG DT/PT CORRESPONDENCE Davide Accadia Università di Trieste, Dipartimento MIGe, Via Valerio 12/1, 34127, Trieste, Italy & Istituto Nazionale di Fisica Nucleare (INFN), Sezion...

work page 1993

[1] [1]

Geom.8(2021), no

Noah Arbesfeld,K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface, Algebr. Geom.8(2021), no. 5, 587–625

work page 2021

[2] [2]

Bessenrodt,On hooks of young diagrams, Annals of Combinatorics2(1998), 103–110

C. Bessenrodt,On hooks of young diagrams, Annals of Combinatorics2(1998), 103–110

work page 1998

[3] [3]

Yalong Cao, Martijn Kool, and Sergej Monavari,K -theoretic DT/PT correspondence for toric Calabi-Yau 4-folds, Commun. Math. Phys.396(2022), no. 1, 225–264

work page 2022

[4] [4]

Ricolfi,Invariants of nested Hilbert and Quot schemes on surfaces, ArXiv:2503.14175, 2025

Nadir Fasola, Michele Graffeo, Danilo Lewa´nski, and Andrea T . Ricolfi,Invariants of nested Hilbert and Quot schemes on surfaces, ArXiv:2503.14175, 2025

work page arXiv 2025

[5] [5]

Gansner,The Hillman-Grassl correspondence and the enumeration of reverse plane partitions, J

Emden R. Gansner,The Hillman-Grassl correspondence and the enumeration of reverse plane partitions, J. Combin. Theory Ser. A30(1981), no. 1, 71–89

work page 1981

[6] [6]

Cruz Godar and Benjamin Young,Bijectivizing the PT-DT correspondence, Electron. J. Comb.32(2025), no. 2, 35

work page 2025

[7] [7]

Ricolfi, and Alessio Sammartano,The geometry of double nested Hilbert schemes of points on curves, Trans

Michele Graffeo, Paolo Lella, Sergej Monavari, Andrea T . Ricolfi, and Alessio Sammartano,The geometry of double nested Hilbert schemes of points on curves, Trans. Am. Math. Soc.378(2025), no. 9, 6013–6047

work page 2025

[8] [8]

Ricolfi,Enumeration of partitions via socle reduction, ArXiv:2501.10267, 2025

Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T . Ricolfi,Enumeration of partitions via socle reduction, ArXiv:2501.10267, 2025

work page arXiv 2025

[9] [9]

Ricolfi,The motive of the Hilbert scheme of points in all dimensions, Proceedings of the London Mathematical Society132(2026), no

Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T . Ricolfi,The motive of the Hilbert scheme of points in all dimensions, Proceedings of the London Mathematical Society132(2026), no. 3, e70140

work page 2026

[10] [10]

A. P . Hillman and R. M. Grassl,Reverse plane partitions and tableau hook numbers, J. Comb. Theory, Ser. A21 (1976), 216–221

work page 1976

[11] [11]

High Energy Phys

Amer Iqbal, Can Kozçaz, and Cumrun Vafa,The refined topological vertex, J. High Energy Phys. (2009), no. 10, 069

work page 2009

[12] [12]

Helen Jenne, Gautam Webb, and Benjamin Young,The combinatorial PT-DT correspondence, Sémin. Lothar. Comb.85B(2021), 12, Id/No 89

work page 2021

[13] [13]

Nikolas Kuhn, Henry Liu, and Felix Thimm,Wall-crossing for invariants of equivariant 3CY categories, ArXiv:2512.23012, 2025

work page arXiv 2025

[14] [14]

Topol.30(2026), no

Nikolas Kuhn, Henry Liu, and Felix Thimm,The 3-fold K-theoretic DT/PT vertex correspondence holds, Geom. Topol.30(2026), no. 1, 71–154

work page 2026

[15] [15]

Math.158 (2022), no

Sergej Monavari,Double nested Hilbert schemes and the local stable pairs theory of curves, Compos. Math.158 (2022), no. 9, 1799–1849

work page 2022

[16] [16]

Sergej Monavari,The refined local Donaldson–Thomas theory of curves, ArXiv:2506.14359, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025

[17] [17]

18, American Mathematical Society, Providence, RI, 1999

Hiraku Nakajima,Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999

work page 1999

[18] [18]

Andrei Okounkov and Rahul Pandharipande,Gromov-Witten theory, Hurwitz theory, and completed cycles, Annals of mathematics (2006), 517–560

work page 2006

[19] [19]

Tullio Levi-Civita

Bruce E. Sagan,Combinatorial proofs of hook generating functions for skew plane partitions, Theor. Comput. Sci.117(1993), no. 1-2, 273–287. 20 ON THE COMBINATORICS OF THE REFINED 1-LEG DT/PT CORRESPONDENCE Davide Accadia Università di Trieste, Dipartimento MIGe, Via Valerio 12/1, 34127, Trieste, Italy & Istituto Nazionale di Fisica Nucleare (INFN), Sezion...

work page 1993