At most $3.55^n$ stable matchings

TL;DR

This paper significantly improves the upper bound on the number of stable matchings for equal numbers of jobs and applicants, from an exponential of 131072^n to 3.55^n, using a new entropy-based formulation.

Contribution

It introduces a novel entropy formulation that simplifies bounding the maximum number of stable matchings, advancing combinatorial counting methods.

Findings

Upper bound on stable matchings improved to 3.55^n

New entropy-based formulation for counting combinatorial objects

Potential applicability to other combinatorial counting problems

Abstract

We improve the upper bound for the maximum possible number of stable matchings among $n$ jobs and $n$ applicants from $131072^n+O(1)$ to $3.55^n+O(1)$. To establish this bound, we state a novel formulation of a certain entropy bound that is easy to apply and may be of independent interest in counting other combinatorial objects