Pattern Avoidance in Partial Words Dense with Holes.

Abstract

A \textit{partial word} is a sequence of symbols over a finite alphabet that may have some undefined positions, called \textit{holes}, that match every letter of the alphabet. Previous work completed the classification of all binary patterns with respect to partial word avoidability. In this paper, we pose the problem of avoiding patterns in partial words very dense with holes. We define the concept of hole sparsity, a measure of the frequency of holes in a partial word, and determine the minimum hole sparsity for all unary patterns in the context of trivial and non-trivial avoidability. Results for more general patterns are also given. Furthermore, we discuss hole spacing and hole density for abelian powers.