Automated Counting of Restricted Motzkin Paths

TL;DR

This paper introduces two automated methods, numeric and symbolic dynamic programming, for enumerating restricted Motzkin paths with various height and run-length constraints, implemented in Maple packages.

Contribution

It presents novel automated algorithms for enumerating complex families of restricted Motzkin paths, enhancing computational efficiency and capability.

Findings

Numeric dynamic programming can be slow for large problems.

Symbolic dynamic programming effectively handles larger path families.

Methods are implemented in accessible Maple packages.

Abstract

Motzkin paths are simple yet important combinatorial objects. In this paper, we consider families of Motzkin paths with restrictions on peak heights, valley heights, upward-run lengths, downward-run lengths, and flat-run lengths. This paper presents two fully automated methods for enumerating the paths of such families. The first method uses numeric dynamic programming. While this method often times works, it can be slow and may not work for larger problems. The second method uses symbolic dynamic programming to solve such problems. These methods are implemented in the maple packages accompanying this article.