Combinatorics of lattice paths with and without spikes

We derive a series of results on random walks on a d -dimensional hypercubic lattice (lattice paths). We introduce the notions of terse and simple paths corresponding to the path having no backtracking parts (spikes). These paths label equivalence classes which allow a rearrangement of the sum over paths. The basic combinatorial quantities of this construction are given. These formulae are useful when performing strong-coupling (hopping parameter) expansions of lattice models. Some applications are described.