Counting Bordered Partial Words by Critical Positions

A partial word, sequence over a finite alphabet that may have some undefined positions or holes, is bordered if one of its proper prefixes is compatible with one of its suffixes. The number theoretical problem of enumerating all bordered full words (the ones without holes) of a fixed length n over an alphabet of a fixed size k is well known. In this paper, we investigate the number of bordered partial words having h holes with the parameters k, n. It turns out that all borders of a full word are simple, and so every bordered full word has a unique minimal ∗This material is based upon work supported by the National Science Foundation under Grants DMS–0452020 and DMS–0754154. The Department of Defense is also gratefully acknowledged. This work was done during the last author’s stay at the University of North Carolina at Greensboro. A preliminary version of this paper was orally presented under the title “Counting Distinct Partial Words” at the International Conference on Automata, Languages and Related Topics that was held in Debrecen, Hungary in October 2008, and a one-page abstract appeared in the proceedings of the conference. We thank the referee of a preliminary version of this paper for his/her very valuable comments and suggestions. Department of Mathematical Sciences, Carnegie Mellon University, 5032 Forbes Ave., Pittsburgh, PA 15289, USA Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 27402–6170, USA, [email protected] Department of Mathematics, University of Mississippi, P.O. Box 1848, University, MS 38677, USA Department of Mathematics, Princeton University, Washington Road, Princeton, NJ 08544–1000, USA GRLMC, Universitat Rovira i Virgili, Campus Catalunya, Departament de Filologies Romàniques, Av. Catalunya, 35, Tarragona, 43002, Spain