Automatic Counting of Restricted Dyck Paths via (Numeric and Symbolic) Dynamic Programming

TL;DR

This paper introduces two automated methods, numeric and symbolic dynamic programming, for counting restricted Dyck paths, enhancing efficiency and automation in combinatorial enumeration tasks.

Contribution

It presents fully automated numeric and symbolic dynamic programming approaches implemented in Maple for counting various families of Dyck paths.

Findings

Numeric dynamic programming effectively counts smaller Dyck path families.

Symbolic dynamic programming handles larger, more complex Dyck path problems.

Both methods are fully automated and implemented in Maple.

Abstract

Dyck paths are one of the most important objects in enumerative combinatorics, and there are many papers devoted to counting selected families of Dyck paths. Here we present two approaches for the automatic counting of many such families, using both a "dumb" approach (driven by numeric dynamic programming) that often works in practice, and a "clever" approach, needed for larger problems, driven by "symbolic" dynamic programming. Both approaches are fully automated and implemented in Maple.