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Generalized Dyck paths. (English) Zbl 0723.05074
Let $${\mathcal S}$$ be a finite multi-set (a set with repetitions) of vectors in $${\mathbb{N}}\times {\mathbb{N}}$$ and let $${\mathcal S}^*=\{(r,-s)+(r,s)\in {\mathcal S}\}.$$ An A-path ($${\mathcal S}$$-Dyck path) is a path in $${\mathbb{Z}}\times {\mathbb{Z}}$$ which starts from (0,0) and ends on the x-axis, uses only vectors from $${\mathcal S}+{\mathcal S}^*$$ and never goes below the x-axis. The author defines five other related types of paths. Each type gives rise to a generating function of several variables. These functions satisfy a system of equations which yield a polynomial equation satisfied by A, the generating function for A-paths. For example, when $${\mathcal S}=\{{\mathbf{u}}_ 1,{\mathbf{u}}_ 2\}$$ where u$${}_ 1=(r_ 1,1)$$, u$${}_ 2=(r_ 2,2)$$. A satisfies a 4th degree polynomial equation.

##### MSC:
 05C38 Paths and cycles
##### Keywords:
Dyck path; Catalan number; generating function
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##### References:
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