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 dpuqdr for any positive integers p,qp,q, and rr. We find that both numbers of types dpudr and dpu2+dr are independent of the number of ii flaws for 1≤i≤n−11≤i≤n−1, i.e., they satisfy the Chung–Feller property, where u is a unit step, uk is an up stepstep of length kk, and u2+=us1us2⋯ust with ∑i=1tsi≥2. The enumeration of generalized lattice paths by peaks and by ascents is also studied.