The relevant prefixes of coloured Motzkin walks: An average case analysis

In this paper we study some relevant prefixes of coloured Motzkin walks (otherwise called coloured Motzkin words). In these walks, the three kinds of step can have , and colours, respectively. In particular, when = = = 1 we have the classical Motzkin walks while for = = 1 and = 0 we find the well-known Dyck walks. By using the concept of Riordan arrays and probability generating functions we find the average length of the relevant prefix in a walk of length n and the corresponding variance in terms of , and . This result is interesting from a combinatorial point of view and also provides an average case analysis of algorithms related to the problem of ranking and generating uniformly at random the coloured Motzkin words.

• Type: Journal
• Year: 2010
• Where: TCS
• Authors: Donatella Merlini, Renzo Sprugnoli
• Keywords: Average Case Analysis, Classical Motzkin, Motzkin Words, TCS 2010, Theoretical Computer Science