Experimenting with the Garsia-Milne Involution Principle

TL;DR

This note reframes the Garsia–Milne involution principle in a broad probabilistic setting, moving beyond partition identities to a random matching model with $c$ cheating and $f$ faithful agents. It derives exact distributions for both the number of requests by a single faithful participant and the total number of requests, via hypergeometric and related generating-function expressions, and analyzes their asymptotic geometric limits. The authors implement these ideas in a Maple-based GMIP.txt package and accompanying tools RandL and GMstory to simulate and illustrate the theory. Overall, the work highlights the versatility of the involution principle and demonstrates how probabilistic analysis can illuminate bijective proof techniques in combinatorics.

Abstract

In 1981, Adriano Garsia and Steve Milne found the first bijective proof of the celebrated Rogers-Ramanujan identities. To achieve this feat, they invented a versatile tool that they called the Involution Principle. In this note we revisit this useful principle from a very general perspective, independent of its application to specific combinatorial identities, and will explore its complexity.