On the Theory of Pfaffian Orientations. I. Perfect Matchings and Permanents

Anna Galluccio, Martin Loebl


Kasteleyn stated that the generating function of the perfect matchings of a graph of genus $g$ may be written as a linear combination of $4^g$ Pfaffians. Here we prove this statement. As a consequence we present a combinatorial way to compute the permanent of a square matrix.

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