Using Symbolic Computation to Explore Generalized Dyck Paths and Their Areas

TL;DR

This paper demonstrates how to use symbolic computation and Groebner Bases to automatically derive algebraic equations for generating functions of generalized Dyck paths and compute statistical properties of their areas.

Contribution

It introduces a novel combination of Groebner Basis algorithms with symbolic dynamical programming and calculus to analyze generalized Dyck paths and their area statistics.

Findings

Automated derivation of algebraic equations for generating functions.

Calculation of area sums and moments for generalized Dyck paths.

Enabling statistical analysis of path areas using symbolic computation.

Abstract

We show the power of Bruno Buchberger's seminal Groebner Basis algorithm, interfaced, seamlessly, with what we call symbolic dynamical programming, to automatically generate algebraic equations satisfied by the generating functions enumerating so-called Generalized Dyck Walks, i.e. 2D walks that start and end on the x-axis, and never dip below it, for an arbitrary set of steps. More impressively, we combine it with calculus (that Maple knows very well!), to automatically compute generating functions for the sum-of-the-areas of these generalized Dyck paths, and even for the sum of any given power of the areas, enabling us to get statistical information about the area under a random generalized Dyck path.