  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
Lindstr\"om-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 1990). Our results, which
are applicable to graphs with cycles, are formulated in terms of systems of
nonintersecting paths and nonintersecting cycles in digraphs.
