Periodicity in Rectangular Arrays

TL;DR

This paper extends the Lyndon-Schutzenberger periodicity theorem to two-dimensional arrays, introduces the concept of primitive arrays, counts their number, and provides linear-time algorithms for testing primitivity and finding primitive roots.

Contribution

It generalizes periodicity concepts to 2D arrays, introduces primitive array counting, and develops efficient algorithms for primitivity testing and primitive root computation.

Findings

Counted the number of primitive m x n arrays.

Developed linear-time algorithms for primitivity testing.

Extended periodicity theorems to two-dimensional arrays.

Abstract

We discuss several two-dimensional generalizations of the familiar Lyndon-Schutzenberger periodicity theorem for words. We consider the notion of primitive array (as one that cannot be expressed as the repetition of smaller arrays). We count the number of m x n arrays that are primitive. Finally, we show that one can test primitivity and compute the primitive root of an array in linear time.