How many perfect matchings are contained in a given bipartite graph?
An exercise in Godsil's 1993 \textit{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
\textit{An Introduction to Combinatorial Analysis}.
It was stated more explicitly and proved independently by
S.A.~Joni and G.-C.~Rota [\textit{JCTA} \textbf{29} (1980),
59--73] and C.D.~Godsil [\textit{Combinatorica} \textbf{1}
(1981), 257--262]. Another generation later,
perhaps it's time both to simplify the proof and to broaden the
formula's reach