On directed lattice paths with vertical steps

This paper is devoted to the study of non-simple directed lattice paths running between two fixed points and for which the set of allowed steps contains vertical step V=(0,−1)V=(0,−1) and forward steps Sk=(1,k)Sk=(1,k) for some k∈Zk∈Z. These paths generalize the heavily-studied simple directed lattice paths that consist of only forward steps. Two special families of primary (restricted to the half-plane) and free (unrestricted) lattice paths are considered. It is shown that for any family of primary paths with vertical steps there is equinumerous family of proper weighted simple directed lattice paths. The relationship between primary and free paths is established and some combinatorial and statistical properties are obtained. Finally, four families of paths with vertical steps are presented and related to Łukasiewicz, Raney, Dyck, Motzkin, Schröder, and Delannoy paths.