Counting Lattice Paths via a New Cycle Lemma

Let $\alpha,\beta,m,n$ be positive integers. Fix a line $L:y=\alpha x+\beta$ and a lattice point $Q=(m,n)$ on L. It is well known that the number of lattice paths from the origin to Q which touch L only at Q is given by $\frac{\beta}{m+n}\binom{m+n}m.$ We extend the above formula in various ways; in particular, we consider the case when $\alpha$ and $\beta$ are arbitrary positive reals. The key ingredient of our proof is a new variant of the cycle lemma originated by Dvoretzky and Motzkin [Duke Math. J., 14 (1947), pp. 305–313] and Raney [Trans. Amer. Math. Soc., 94 (1960), pp. 441–451]. We also include a counting formula for lattice paths lying under a cyclically shifting boundary, which generalizes a resultdue to Irving and Rattan in [J. Combin. Theory Ser. A, 116 (2009), pp. 499–514], and a counting formula for lattice paths having a given number of peaks, which contains the Narayana number as a special case.