# Symmetries of shamrocks IV: The self-complementary case

## TL;DR

This paper enumerates centrally symmetric lozenge tilings of a hexagon with a shamrock removed, introduces a novel proof technique, and applies it to self-complementary plane partitions, advancing combinatorial enumeration methods.

## Contribution

It presents a new enumeration of symmetric tilings using a modified Kuo's condensation and provides a novel proof for self-complementary plane partitions.

## Key findings

- Enumeration of centrally symmetric lozenge tilings with shamrock removal
- Development of a variant of Kuo's graphical condensation
- New proof for self-complementary plane partitions

## Abstract

In this paper we enumerate the centrally symmetric lozenge tilings of a hexagon with a shamrock removed from its center. Our proof is based on a variant of Kuo's graphical condensation method in which only three of the four involved vertices are on the same face. As a special case, we obtain a new proof of the enumeration of the self-complementary plane partitions.