An extension of Pólya's enumeration theorem

Abstract

In combinatorics, Pólya's Enumeration Theorem is a powerful tool for solving a wide range of counting problems, including the enumeration of groups, graphs, and chemical compounds. In this paper, we present an extension of Pólya's Enumeration Theorem. As an application, we derive a formula that expresses the n-th elementary symmetric polynomial in m indeterminates (where $n\le m$) as a variant of the cycle index polynomial of the symmetric group $\mathrm{Sym}\left(n\right)$. This result resolves a problem posed by Amdeberhan in 2012.