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

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 revisit the theorem and to broaden the formula’s reach.