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arXiv · 2108.03037

On the generating functions of pattern-avoiding Motzkin paths

Christian Bean·Antonio Bernini·Matteo Cervetti·Luca Ferrari
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Abstract

Using a recursive approach, we show that the generating function for sets of Motzkin paths avoiding a single (not necessarily consecutive) pattern is rational over $x$ and the Catalan generating function $C(x) = \frac{1-\sqrt{1-4x^2}}{2x^2}$, where $x$ keeps track of the length of the path. Moreover, an algorithm is provided for finding the generating function in the more general case of an arbitrary set of patterns. In addition, this algorithm allows us to find a combinatorial specification for pattern-avoiding Motzkin paths, which can be used not only for enumeration, but also for exhaustive and random generation.

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Christian Bean, Antonio Bernini, Matteo Cervetti, Luca Ferrari. 2021-08-06. On the generating functions of pattern-avoiding Motzkin paths. https://arxiv.org/abs/2108.03037

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