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We show that a formalism proposed by Creutz to evaluate Grassmann integralsprovides an algorithm of complexity $O(2^n n^3)$ to compute the generatingfunction for the sum of the permanental minors of a matrix of order $n$. Thisalgorithm improves over the Brualdi-Ryser formula, whose complexity is at least$O(2^{\frac{5n}{2}})$. In the case of a banded matrix with band width $w$ andrank $n$ the complexity is $O(2^{min(2w, n)} (w + 1) n^2)$. Related algorithmsfor the matching and independence polynomials of graphs are presented.

Sums of permanental minors using Grassmann algebra