On the generating functions of pattern-avoiding Motzkin paths

TL;DR

This paper develops a recursive method to derive rational generating functions for Motzkin paths avoiding specific patterns, providing algorithms for enumeration, specification, and random generation of such paths.

Contribution

It introduces a recursive approach to find rational generating functions for pattern-avoiding Motzkin paths and offers an algorithm for arbitrary pattern sets.

Findings

Generating functions are rational over x and Catalan function for pattern-avoiding paths.

An algorithm for arbitrary pattern sets is provided.

The method enables enumeration, exhaustive, and random generation of paths.

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.