On the Parikh-de-Bruijn grid

TL;DR

This paper introduces the Parikh-de-Bruijn grid, a graph structure that provides new insights into Parikh vectors and strings, and explores its applications to Parikh-de-Bruijn strings, the abelian analogs of de-Bruijn sequences.

Contribution

It defines the Parikh-de-Bruijn grid and demonstrates its usefulness in analyzing Parikh vectors and strings, including new results on Parikh-de-Bruijn strings.

Findings

Introduces the Parikh-de-Bruijn grid as a new graph structure.

Provides structural insights into Parikh vectors and sets.

Proves results on Parikh-de-Bruijn strings, abelian analogs of de-Bruijn sequences.

Abstract

We introduce the Parikh-de-Bruijn grid, a graph whose vertices are fixed-order Parikh vectors, and whose edges are given by a simple shift operation. This graph gives structural insight into the nature of sets of Parikh vectors as well as that of the Parikh set of a given string. We show its utility by proving some results on Parikh-de-Bruijn strings, the abelian analog of de-Bruijn sequences.