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    Counting generalized Dyck paths

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    The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from (0,0)(0,0) to (n,n)(n,n) which is below the diagonal line y=xy=x. One way to generalize the definition of Dyck path is to change the end point of Dyck path, i.e. we define (generalized) Dyck path to be a lattice path from (0,0)(0,0) to (m,n)∈N2(m,n) \in \mathbb{N}^2 which is below the diagonal line y=nmxy=\frac{n}{m}x, and denote by C(m,n)C(m,n) the number of Dyck paths from (0,0)(0,0) to (m,n)(m,n). In this paper, we give a formula to calculate C(m,n)C(m,n) for arbitrary mm and nn.Comment: 15 pages, 2 figure
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