Plane partitions II: 5 1/2 symmetry classes

TL;DR

This paper provides simplified proofs for counting five symmetry classes of plane partitions within a box, using determinant evaluations and combinatorial arguments, improving upon more complex previous proofs.

Contribution

It introduces new, elementary proofs for five symmetry classes of plane partitions, simplifying enumeration methods and determinant evaluations.

Findings

Simplified enumeration proofs for five symmetry classes

Elementary proof for a specific symmetry class with equal sides

Explicit evaluations of complex determinants related to plane partitions

Abstract

We present new, simple proofs for the enumeration of five of the ten symmetry classes of plane partitions contained in a given box. Four of them are derived from a simple determinant evaluation, using combinatorial arguments. The previous proofs of these four cases were quite complicated. For one more symmetry class we give an elementary proof in the case when two of the sides of the box are equal. Our results include simple evaluations of the determinants $\det(\delta_{ij}+{x+i+j\choose i})_{0\leq i,j\leq n-1}$ and $\det({x+i+j\choose 2j-i})_{0\leq i,j\leq n-1}$, notorious in plane partition enumeration, whose previous evaluations were quite intricate.