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

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.