Counting Lattice Paths via a New Cycle Lemma

Let α, β,m, n be positive integers. Fix a line L : y = αx + 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 m ” . 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 [1] and Raney [8]. 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 [6], and a counting formula for lattice paths having given number of peaks, which contains the Narayana number as a special case.