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

TL;DR

This paper offers new proofs and formulas related to plane partitions, connecting combinatorics with geometric DT/PT wallcrossing, and interprets results within Fock space formalism.

Contribution

It introduces novel proofs, closed formulas, and identities for plane partitions, linking combinatorics with geometric wallcrossing and Fock space representations.

Findings

New proof of Bessenrodt's relation among generating series

Established closed formulas for weighted enumeration of plane partitions

Derived identities in Fock space via bosonic/fermionic formalism

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.