On the generating functions of pattern-avoiding Motzkin paths. (November 2022)

• Record Type: Journal Article
• Authors: Bean, Christian; Bernini, Antonio; Cervetti, Matteo; Ferrari, Luca
• Is Part Of: Journal of symbolic computation. Volume 113(2022)
• Page Start: 126
• Page End: 138
• Publication Date: 2022-11
• Subjects: 05A10 -- 05A15; Lattice path -- Generating function -- Motzkin path -- Pattern -- Combinatorial specification; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2022.02.006
• Languages: English
• ISSNs: 0747-7171

Abstract

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 ) = 1 − 1 − 4 x 2 2 x 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.