A generating function approach to counting theorems for square-free polynomials and maximal tori

TL;DR

This paper introduces generating function techniques to provide elementary proofs and extensions of enumerative results related to square-free polynomials and maximal tori, originally proved using advanced topology and representation theory.

Contribution

It offers a simpler, generating function-based approach to key theorems on square-free polynomials and maximal tori, extending previous results.

Findings

Elementary proofs of existing theorems

Extensions to previous enumerative results

Simplification of complex topological methods

Abstract

A recent paper of Church, Ellenberg, and Farb uses topology and representation theory of the symmetric group to prove enumerative results about square-free polynomials and F-stable maximal tori of the general linear group over the algebraic closure of F_q. In this note, we use generating functions to give elementary proofs of some of their results, and some extensions.