Enumeration of generalized lattice paths by string types, peaks, and ascents

We consider lattice paths with arbitrary step sizes, called generalized lattice paths, and we enumerate them with respect to string types of d(p)u(q)d(r) for any positive integers p, q, and r. We find that both numbers of types d(p)ud(r) and d(p)u(2+)d(r) are independent of the number of i flaws for 1 <= i <= n - 1, i.e., they satisfy the Chung-Feller property, where u is a unit step, u(k) is an up step of length k, and u(2+) = u(s1)u(S2) ... u(st) with Sigma(t)(i=1) S-i >= 2. The enumeration of generalized lattice paths by peaks and by ascents is also studied. (C) 2016 Published by Elsevier B.V.