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August 17, 2025

Another short proof of the Joni-Rota-Godsil integral formula for counting bipartite matchings

EEErin E. EmersonUniversity of Massachusetts Amherst PKP. Mark KayllUniversity of Montana

Key Points

The proof shows how many perfect matchings exist in a bipartite graph, building on prior work in combinatorial math.
Key evidence includes a connection to rook polynomials which play a crucial role in counting matchings.
An observational approach indicates that the formula leverages insights from earlier combinatorial theories established by Joni and Rota.
The findings highlight a need for further exploration into the integrals that characterize bipartite matchings in more complex graphs.

Abstract

How many perfect matchings are contained in a given bipartite graph? An exercise in Godsil's 1993 Algebraic Combinatorics solicits proof that this question's answer is an integral involving a certain rook polynomial. Though not widely known, this result appears implicitly in Riordan's 1958 An Introduction to Combinatorial Analysis. It was stated more explicitly and proved independently by S. A. ~Joni and G. -C. ~Rota JCTA 29 (1980), 59--73 and C. D. ~Godsil Combinatorica 1 (1981), 257--262. Another generation later, perhaps it's time both to simplify the proof and to broaden the formula's reach.

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Cite This Study

Emerson et al. (Fri,) studied this question.

synapsesocial.com/papers/68a370e80a429f797333340f https://doi.org/https://doi.org/10.55016/ojs/cdm.v4i2.62044

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