A combinatorial result with applications to self-interacting random walks

We give a series of combinatorial results that can be obtained from any two collections (both indexed by $\mathbb{Z}\times \mathbb{N}$) 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.