Counting lattice paths via a cycle lemma

Let α, β, m, n be positive integers. Fix a line L : y = α x + β , 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 touches L only at Q is given by β m + n ( m + n n ) . We extend the above formula in various ways, in particular, we consider the case when α and β are arbitrary positive reals. The key ingredient of our proof is a new variant of the cycle lemma originated from Dvoretzky–Motzkin [A. Dvoretzky, Th. Motzkin. A problem of arrangements. Duke Math. J., 14 (1947) 305–313] and Raney [G. Raney. Functional composition patterns and power series reversion. Trans. Amer. Math. Soc., 94 (1960) 441–451]. We also include a counting formula for lattice paths lying under a cyclically shifting boundary, which generalizes a result due to Irving and Ratten in [J. Irving, A. Rattan. The number of lattice paths below a cyclically shifting boundary. J. Combin. Theory Ser. A, 116 (2009) 499–514], and a counting formula for lattice paths having given number of peaks, which contains the Narayana number as a special case.