Abstract
After recalling the definition of Grassmann algebra and elements of Grassmann–Berezin calculus, we use the expression of Pfaffians as Grassmann integrals to generalize a series of formulas relating generating functions of paths in digraphs to Pfaffians. We start with the celebrated Lindström–Gessel–Viennot formula, which we derive in the general case of a graph with cycles. We then make further use of Grassmann algebraic tools to prove a generalization of the results of Stembridge [13]. Our results, which are applicable to graphs with cycles, are formulated in terms of systems of nonintersecting paths and nonintersecting cycles in digraphs.
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