How Many Holes Can an Unbordered Partial Word Contain?

Abstract

Partial words are sequences over a finite alphabet that may have some undefined positions, or “holes,” that are denoted by $\ensuremath{\diamond}$’s. A nonempty partial word is called bordered if one of its proper prefixes is compatible with one of its suffixes (here $\ensuremath{\diamond}$ is compatible with every letter in the alphabet); it is called unbordered otherwise. In this paper, we investigate the problem of computing the maximum number of holes a partial word of a fixed length can have and still fail to be bordered.

This material is based upon work supported by the National Science Foundation under Grant No. DMS–0754154. We thank the referees of a preliminary version of this paper for their very valuable comments and suggestions. A World Wide Web server interface has been established at www.uncg.edu/cmp/research/countingpwords for automated use of the program. This work was done during the fourth author’s stay at the University of North Carolina at Greensboro.