@John Thank you very much for the hints in [comment #83](https://forum.azimuthproject.org/discussion/comment/19032/#Comment_19032)!
Unfortunately, I cannot see what choice of quantale \$$\mathcal{V} = (V, \le, I, \otimes) \$$ allows us compute the number of paths of length \$$n \$$.
The formula to multiply two \$$\mathcal{V} \$$-matrices \$$M \$$ and \$$N \$$ (equation 2.97 in the book) is:
\$(M * N)(x, z) := \bigvee_{y \in Y} M(x, y) \otimes N(y, z). \$
It seems to me that in order to compute the number of paths of length \$$n \$$ we need a quantale whose multiplication corresponds to matrix multiplication:
\$(M * N)(x, z) := \sum_{y \in Y} M(x, y) * N(y, z). \$