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.

Figures (7)

• Figure 1: Respectively from the left, a Young diagram of size 8, a reversed plane partition of size 21 and a skew plane partition of size 23.
• Figure 2: Let $\lambda = (8,4,3,2,2)$. The number of $\diamondsuit$ on the right (resp. of $\heartsuit$ below) of $\Box$ is the arm (resp. leg) length of the hook defined by $\Box$: $a_{\lambda}(\Box) = 2,\, \ell_{\lambda}(\Box) = 3,\, h_{\lambda}(\Box) = 6$.
• Figure 3: The partition $\lambda = (24,11,5^3,1^6)$ on the left is thin, whereas the partition $\mu = (12,10,8^6,1^3)$ on the right is not.
• Figure 4: Example of tectonic movement.
• Figure 5: Procedure for a thick partition.

Theorems & Definitions (32)

• Theorem 1.1: Bessenrodt
• Theorem 1.2: \ref{['theorem:hooks:strips']}
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5: \ref{['cor:hook:strip:refined']}
• Proposition 1.6: \ref{['prop: ultimate wall refined']}
• Definition 2.1
• Proposition 2.2
• proof
• Remark 2.3