(-1)-enumeration of self-complementary plane partitions

TL;DR

This paper establishes a product formula for counting certain self-complementary plane partitions with a specific weight change property, using lattice paths and Pfaffian techniques to connect to known enumeration results.

Contribution

It introduces a new enumeration formula for weighted self-complementary plane partitions involving sign-changing orbits, expanding the combinatorial understanding of these objects.

Findings

Derived a product formula for weighted enumeration

Expressed enumeration as a Pfaffian using lattice paths

Connected new enumeration to known counts of self-complementary plane partitions

Abstract

We prove a product formula for the remaining cases of the weighted enumeration of self-complementary plane partitions contained in a given box where adding one half of an orbit of cubes and removing the other half of the orbit changes the sign of the weight. We use nonintersecting lattice path families to express this enumeration as a Pfaffian which can be expressed in terms of the known ordinary enumeration of self-complementary plane partitions.