An Extension of P\'{o}lya's Enumeration Theorem

TL;DR

This paper extends Pólya's Enumeration Theorem to derive a new formula for elementary symmetric polynomials, addressing a problem posed by Amdeberhan, and broadens its applications in combinatorics.

Contribution

It introduces an extension of Pólya's Enumeration Theorem and provides a novel formula for elementary symmetric polynomials as a variant of the cycle index polynomial.

Findings

Derived a new formula for elementary symmetric polynomials

Resolved a problem posed by Amdeberhan in 2012

Extended the applicability of Pólya's Enumeration Theorem

Abstract

In combinatorics, P\'{o}lya'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 P\'{o}lya's Enumeration Theorem. As an application, we derive a formula that expresses the $n$-th elementary symmetric polynomial in $m$ indeterminates (where $n\leq m$) as a variant of the cycle index polynomial of the symmetric group $\mathrm{Sym}(n)$. This result resolves a problem posed by Amdeberhan in 2012.