On the structure of sequences with minimal maximal pattern complexity

TL;DR

This paper investigates the structure of sequences with minimal maximal pattern complexity over larger alphabets, characterizing their properties and establishing exact complexity bounds for aperiodic sequences.

Contribution

It provides a detailed analysis of low maximal pattern complexity sequences over ferent alphabets and determines the minimal complexity for aperiodic sequences using all letters.

Findings

Minimal maximal pattern complexity for aperiodic sequences is 2k + ferent alphabet size - 2.

Sequences with this minimal complexity have a specific, characterized structure.

Sequences with complexity less than this bound must have some periodic structure.

Abstract

In 2002, Kamae and Zamboni introduced maximal pattern complexity and determined that any aperiodic sequence must have maximal pattern complexity at least $2k$. In 2006, Kamae and Rao examined the maximal pattern complexity of sequences over larger alphabets and showed that sequences which have maximal pattern complexity less than $\ell k$, for $\ell$ the size of the alphabet, must have some periodic structure. In this paper, we investigate the structure of sequences of low maximal pattern complexity over $\ell$ letters where $\liminf\limits\limits_{k \to \infty} p_{\alpha}^*(k) - 3k = -\infty$. In addition, we show that the minimal maximal pattern complexity of an aperiodic sequence which uses all $\ell$ letters is $p_{\alpha}^*(k) = 2k + \ell -2$, and give an exact structure for aperiodic sequences with this maximal pattern complexity.