Equivalence classes of nodes in trees and rational generating functions

TL;DR

This paper introduces a criterion for establishing the rationality of generating functions for node counts in trees and applies it to various combinatorial and algebraic problems, revealing new connections and classical results.

Contribution

It presents a novel criterion for proving the rationality of generating functions and demonstrates its applications across group theory, combinatorics, and finite vector spaces.

Findings

Rational generating functions for node counts can be computed using the criterion.

Connections between counting vector configurations and Gaussian binomial coefficients are established.

Classical combinatorial results like Bell and Stirling numbers are derived from the method.

Abstract

Let $c_n$ denote the number of nodes at a distance $n$ from the root of a rooted tree. A criterion for proving the rationality and computing the rational generating function of the sequence $\{c_n\}$ is described. This criterion is applied to counting the number of conjugacy classes of commuting tuples in finite groups and the number of isomorphism classes of representations of polynomial algebras over finite fields. The method for computing the rational generating functions, when applied to the study of point configurations in finite sets, gives rise to some classical combinatorial results on Bell numbers and Stirling numbers of the second kind. When applied to the study of vector configurations in a finite vector space, it reveals a connection between counting such configurations and Gaussian binomial coefficients.