On the structure of sequences with minimal maximal pattern complexity

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.