Lower Bounds for the Number of Repetitions in 2D Strings

TL;DR

This paper establishes lower bounds on the number of repetitions in 2D strings, demonstrating that the counts of distinct tandems, quartics, and runs can grow faster than previously conjectured, thus advancing understanding of 2D string combinatorics.

Contribution

It constructs infinite families of 2D strings with super-quadratic counts of repetitions, resolving an open question about their maximum possible numbers.

Findings

Constructed 2D strings with BDistinct tandems

Constructed 2D strings with BDistinct quartics

Constructed 2D strings with Bruns

Abstract

A two-dimensional string is simply a two-dimensional array. We continue the study of the combinatorial properties of repetitions in such strings over the binary alphabet, namely the number of distinct tandems, distinct quartics, and runs. First, we construct an infinite family of $n\times n$ 2D strings with $\Omega(n^{3})$ distinct tandems. Second, we construct an infinite family of $n\times n$ 2D strings with $\Omega(n^{2}\log n)$ distinct quartics. Third, we construct an infinite family of $n\times n$ 2D strings with $\Omega(n^{2}\log n)$ runs. This resolves an open question of Charalampopoulos, Radoszewski, Rytter, Wale\'n, and Zuba [ESA 2020], who asked if the number of distinct quartics and runs in an $n\times n$ 2D string is $\mathcal{O}(n^{2})$.