Burnside’s Lemma and Its Applications in Combinatorics Problems

Abstract

The problems of discriminating the faces of a die using several colors and finding the number of ways a necklace can form are extremely famous in combinational analysis. The Burnside’s lemma, which is also called Burnside’s counting theorem or other names, is usually necessary in taking account of symmetries when enumerating nonequivalent patterns. In this paper, starting from the definition and clarifications of important concepts used in the proof of Burnside’s lemma, it presents a clear proof to this significant formula used across combinatorics, group theory, recreational and contest mathematics. Moreover, these definitions are explained in simpler terms, which provide better intuition to the application of the Burnside’s lemma. Next, some famous examples of the application of this lemma are discussed. A simple coloring problem and a simple necklace problem was given in the start of the corresponding sections, allowing readers to understand how this lemma is applicable in these scenarios. Afterwards, Burnside’s lemma is applied to solve more general problems, which are more significant.