Pfaffians and nonintersecting paths in graphs with cycles: Grassmann   algebra methods
  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. 
Sylvain Carrozza and Adrian Tanasa
math.CO hep-th