  We give a series of combinatorial results that can be obtained from any two
collections (both indexed by $\Z\times \N$) of left and right pointing arrows
that satisfy some natural relationship. When applied to certain
self-interacting random walk couplings, these allow us to reprove some known
transience and recurrence results for some simple models. We also obtain new
results for one-dimensional multi-excited random walks and for random walks in
random environments in all dimensions.
