  Walks in a directed graph can be given a partially ordered structure that
extends to possibly unconnected objects, called hikes. Studying the incidence
algebra on this poset reveals unsuspected relations between walks and
self-avoiding hikes. These relations are derived by considering truncated
versions of the characteristic polynomial of the weighted adjacency matrix,
resulting in a collection of matrices whose entries enumerate the self-avoiding
hikes of length $\ell$ from one vertex to another.
