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

On the Combinatorics of the Refined 1-leg DT/PT Correspondence

Introduction and Context

This work provides a detailed combinatorial analysis of the refined 1-leg Donaldson-Thomas/Pandharipande-Thomas (DT/PT) correspondence, with a specific focus on the interaction between the combinatorics of plane partitions, particularly reversed and skew plane partitions, and the wall-crossing phenomena of enumerative invariants for local curves. The refined DT/PT correspondence is a central conjecture/statement in enumerative theory on Calabi-Yau threefolds, capturing the relationship between counting ideal sheaves (DT invariants) and stable pairs (PT invariants), both at the level of generating functions and their refined or K-theoretic analogues.

Traditionally, the equivalence of generating series for DT and PT invariants, and more generally, wall-crossing in their enumerative invariants, is formulated in terms of generating functions for certain 3D or skew partitions, with a rich combinatorial framework involving Young diagrams, (reversed/skew) plane partitions and associated hook-length statistics. This paper approaches these correspondences with new combinatorial proofs, introduces new closed formulae, and develops a canonical Fock space interpretation.

Key Results

Alternative Proof of the Bessenrodt Theorem for Thin Partitions

The authors present a new proof of the Bessenrodt bijection, which relates generating series of reversed plane partitions and skew plane partitions, motivated by wall-crossing formulas for refined DT/PT invariants. A core object is the set of internal and external hooks of a Young diagram $\lambda$ and their enumeration by hook length, arm, and leg statistics.

They introduce the class of thin partitions (Definition 1.1 in the original manuscript), for which they construct an explicit bijection---the so-called "tectonic movement" of Young diagrams—which reorganizes the rectangles (or 'plates') into a configuration where the bijection with external and internal hooks becomes transparent. This provides a conceptually and technically new proof method compared to Bessenrodt's Maya diagram machinery.

Refined Enumerative Identities and Wall-Crossing

By plethystic exponentiation of the basic bijection, the authors obtain refined wall-crossing identities at the level of generating functions, including explicit formulas involving arm and leg lengths of hooks:

$$\prod_{\Box \in \mathcal{H}'(\lambda)} \frac{1}{1 - x^{a(\Box) + 1} y^{\ell(\Box)}} = \prod_{\Box \in \mathcal{H}'(\varnothing)} \frac{1}{1 - x^{a(\Box) + 1}y^{\ell(\Box)}} \prod_{\Box \in \mathcal{H}(\lambda)} \frac{1}{1 - x^{a(\Box) + 1}y^{\ell(\Box)}}$$

This recovers, in a refined setting, earlier results of Sagan for specializations $x = y = q$ (where only hook length, as opposed to arm/leg, is recorded). It is emphasized that this refinement does not extend to keeping track of all monomial weights per box—a negative statement supported by explicit counterexamples.

Hook-to-Strip Bijections and Identities

The paper provides a detailed combinatorial bijection ("hook-to-strip") between pairs $(\lambda,h)$, with $h$ an internal hook of length $\ell$ in a size-$d$ partition, and pairs $(\mu,h')$, with $h'$ an external hook of length $\ell$ in a size-$$\prod_{\Box \in \mathcal{H}'(\lambda)} \frac{1}{1 - x^{a(\Box) + 1} y^{\ell(\Box)}} = \prod_{\Box \in \mathcal{H}'(\varnothing)} \frac{1}{1 - x^{a(\Box) + 1}y^{\ell(\Box)}} \prod_{\Box \in \mathcal{H}(\lambda)} \frac{1}{1 - x^{a(\Box) + 1}y^{\ell(\Box)}}$$0 partition, preserving both location and hook type. This yields new refined identities for weighted enumeration by content statistics.

Fock Space Formalism

These combinatorial correspondences are encoded in the bosonic and fermionic Fock space formalism, with the bijections corresponding to operator symmetries. In particular, the plethystic exponential structures, as well as actions of (twisted) free boson operators on partition bases, are shown to encapsulate the enumerative relationships established combinatorially.

Numerical and Structural Claims

• Explicit closed formulas for the weighted enumeration of reversed and skew plane partitions are given, generalizing Gansner's theorem and refining Sagan's hook-product identities.
• Cases where the refined wall-crossing fails at further levels of refinement (i.e., with monomials in a box-by-box variable assignment) are explicitly described.
• An explicit, algorithmic bijection for thin partitions is constructed, contrasting the previous existential proofs.

Implications

Combinatorics and Enumerative Geometry

The results further systematize the connections between plane partitions, Young diagram combinatorics, and the enumerative geometry of Hilbert schemes and moduli spaces of sheaves (DT/PT theory). The development provides a transparent algorithmic framework for understanding DT/PT wall-crossing—a central phenomenon in curve counting—via solely combinatorial data. The extension to refined settings has implications for refined curve counting invariants, vertex models, and related topics in mathematical physics.

Representation-Theoretic Perspective

The Fock space interpretation enriches the intersection of algebraic combinatorics, quantum algebra, and enumerative geometry, providing operator-theoretic perspectives on wall-crossing formulas and enumerative correspondences. This points towards further connections with infinite wedge spaces, symmetric functions, and vertex algebraic structures.

Potential Future Directions

• Extension to higher-leg and higher-rank refined DT/PT correspondences, especially incorporating non-toric geometries or general moduli problems.
• Exploration of refined invariants under broader wall-crossing conditions, especially in motivic and K-theoretic enumerative geometry.
• Deeper analysis of bijections outside the class of thin partitions and generalizations to skew/hybrid combinatorial structures.

Conclusion

This paper establishes new combinatorial frameworks and bijective methods relevant to the refined 1-leg DT/PT correspondence and provides new identities and operator-theoretic interpretations for the weighted enumeration of plane partitions and their generalizations. The work clarifies the refined wall-crossing at the combinatorial level, with precise structural, algorithmic, and Fock space results. This strengthens the interplay between algebraic combinatorics and enumerative geometric wall-crossing phenomena and lays groundwork for further developments in refined enumerative invariants and their representation-theoretic structures.

Reference: D. Accadia, D. Lewanski, S. Monavari, "On the combinatorics of the refined 1-leg DT/PT correspondence," (2603.29435).