@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). \\]