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{\bf Emily Allen, F. Blanchet-Sadri, Cameron Byrum, Mihai Cucuringu and Robert Merca\c s}
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{\bf Counting Bordered Partial Words by Critical Positions}
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A {\em partial word}, sequence over a finite alphabet that may have
some undefined positions or holes, is {\it bordered} if one of its
proper prefixes is {\em 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. It turns out that all borders of a full word
are {\em simple}, and so every bordered full word has a unique minimal
border no longer than half its length. Counting bordered partial words
having $h$ holes with the parameters $k, n$ is made extremely more
difficult by the failure of that combinatorial property since there is
now the possibility of a minimal border that is {\em nonsimple}. Here,
we give recursive formulas based on our approach of the so-called {\em
simple and nonsimple critical positions}.

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