Counting the number of group orbits by marrying the Burnside process with importance sampling

TL;DR

This work addresses approximating the number of orbits $k(\mathcal{X},G)$ under finite group actions when exact counting is intractable. It develops a general counting algorithm that marries the Burnside process with importance sampling, enabling approximate orbit counts via a nested sequence of sets and groups. The method is specialized to estimate the number of conjugacy classes $k(U_n(\mathbb{F}_q))$ of unitriangular groups using pattern groups, with theoretical variance control and demonstrated numerical accuracy up to sizable $n$ and $q$. The results offer a scalable framework for orbit enumeration and contribute practical tools for conjugacy-class enumeration in algebraic groups.

Abstract

This paper introduces a novel and general algorithm for approximately counting the number of orbits under group actions. The method is based on combining the Burnside process and importance sampling. Specializing to unitriangular groups yields an efficient algorithm for estimating the number of conjugacy classes of such groups.

Figures (6)

• Figure 1: Logarithm of true and estimated number of orbits
• Figure 2: Histogram of logarithm of estimated number of orbits for $k=10$
• Figure 3: Plot of $\log_2(k(U_n(\mathbb{F}_2)))$ for $n=1,2,\cdots,32$
• Figure 4: Plot of $\log_2(k(U_n(\mathbb{F}_2)))\slash (n^2)$ for $n=1,2,\cdots,32$
• Figure 5: Plot of $\log_3(k(U_n(\mathbb{F}_3)))$ for $n=1,2,\cdots,32$

Theorems & Definitions (10)

• Conjecture 1.1
• Proposition 3.1
• proof
• Proposition 4.1
• proof
• Theorem 4.1
• Corollary 4.1.1
• Remark
• proof : Proof of Corollary \ref{['Cor4.1']} (based on Theorem \ref{['Thm2']})
• proof : Proof of Theorem \ref{['Thm2']}