Combinatorics of nondeterministic walks of the Dyck and Motzkin type

Abstract : This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. At each step, a nondeterministic walk draws a random set of steps from a predefined set of sets and explores all possible extensions in parallel. We introduce our new model on Dyck steps with the nondeterministic step set {{−1}, {1}, {−1, 1}} and Motzkin steps with the nondeterministic step set {{−1}, {0}, {1}, {−1, 0}, {−1, 1}, {0, 1}, {−1, 0, 1}}. For general lists of step sets and a given length, we express the generating function of nondeterministic walks where at least one of the walks explored in parallel is a bridge (ends at the origin). In the particular cases of Dyck and Motzkin steps, we also compute the asymptotic probability that at least one of those parallel walks is a meander (stays nonnegative) or an excursion (stays nonnegative and ends at the origin). This research is motivated by the study of networks involving encapsulations and decapsulations of protocols. Our results are obtained using generating functions and analytic combinatorics.
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Submitted on : Saturday, December 15, 2018 - 6:37:31 PM
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  • HAL Id : hal-01910727, version 1
  • ARXIV : 1812.06650

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Elie de Panafieu, Mohamed Lamine Lamali, Michael Wallner. Combinatorics of nondeterministic walks of the Dyck and Motzkin type. Workshop on Analytic Algorithmics and Combinatorics, Jan 2019, San Diego, United States. ⟨hal-01910727⟩

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