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arXiv · 1611.04889

Pfaffians and nonintersecting paths in graphs with cycles: Grassmann algebra methods

Sylvain Carrozza·Adrian Tanasa
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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 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.

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Sylvain Carrozza, Adrian Tanasa. 2016-11-15. Pfaffians and nonintersecting paths in graphs with cycles: Grassmann algebra methods. https://doi.org/10.1016/j.aam.2017.09.003

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