An Extension of Pólya's Enumeration Theorem
Xiongfeng Zhan, Xueyi Huang
TL;DR
The paper extends Pólya's Enumeration Theorem by incorporating a weight-preserving Δ-criterion, yielding a unified framework that connects orbit counts with a lifted weight vector \\tilde{\\boldsymbol{w}}. It proves a signed, Δ-weighted cycle-index identity that expresses the $n$-th elementary symmetric polynomial as a signed cycle-index average over the symmetric group, specifically $e_n(\\boldsymbol{w})=\\frac{1}{n!}\\sum_{\\sigma\\in\\mathrm{Sym}(n)}\\mathrm{sgn}(\\sigma) Z(\\sigma,\\tilde{\\boldsymbol{w}})$. This leads to a determinant-trace correspondence: with $t_i=\\mathrm{tr}(L^i)$, one has $\\mathrm{det}(L)=\\frac{1}{n!}\\sum_{\\sigma\\in\\mathrm{Sym}(n)}\\mathrm{sgn}(\\sigma) Z(\\sigma,\\boldsymbol{t})$, thereby solving Amdeberhan's problem. The results bridge group actions, cycle-index polynomials, and classical symmetric functions, and offer a principled route to expressing determinants via cycle-index data.
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 $n\leq m$) 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.