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The main purpose of this paper is to derive generating functions for the numbers of lattice paths running from (0, 0) to any (n, k) in Z × N consisting of four types of steps: horizontal, and sloping L = (−1, 1). These paths generalize the well-known Delannoy paths which consist of steps H, V , and D. Several restrictions are considered. However, we mainly treat with those which will be needed to get the generating function for the numbers R(n, k) of these lattice paths whose points lie in the integer rectangle {(x, y) ∈ N 2 : 0 ≤ x ≤ n, 0 ≤ y ≤ k}. Recurrence relation, generating functions and explicit formulas are given. We show that most of considered numbers define Riordan arrays.

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The paper is devoted to the study of lattice paths that consist of vertical steps (0, −1) and non-vertical steps (1, k) for some k ∈ Z. Two special families of primary and free lattice paths with vertical steps are considered. It is shown that for any family of primary paths there are equinumerous families of proper weighted lattice paths that consist of only non-vertical steps. The relation between primary and free paths is established and some combinatorial and statistical properties are obtained. It is shown that the expected number of vertical steps in a primary path running from (0, 0) to (n, −1) is equal to the number of free paths running from (0, 0) to (n, 0). Enumerative results with generating functions are given. Finally, a few examples of families of paths with vertical steps are presented and related to Lukasiewicz, Motzkin, Dyck and Delannoy paths.

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