Plane partitions of shifted double staircase shape

TL;DR

This paper derives a new product formula for counting shifted plane partitions of a specific shape, using lozenge tilings and Kuo condensation, marking a significant advancement in combinatorial enumeration.

Contribution

It provides the first new family of shapes with a plane partition product formula in many years, expanding the understanding of enumerative combinatorics.

Findings

Derived a product formula for shifted plane partitions of shifted double staircase shape.

Applied lozenge tilings and Kuo condensation techniques in the proof.

Established a new example of shapes with explicit enumeration formulas.

Abstract

We give a product formula for the number of shifted plane partitions of shifted double staircase shape with bounded entries. This is the first new example of a family of shapes with a plane partition product formula in many years. The proof is based on the theory of lozenge tilings; specifically, we apply the "free boundary" Kuo condensation due to Ciucu.