\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf Tero Harju and Dirk Nowotka}
%
%
\medskip
\noindent
%
%
{\bf Bordered Conjugates of Words over Large Alphabets}
%
%
\vskip 5mm
\noindent
%
%
%
%
The border correlation function attaches to every word $w$ a binary
word $\beta(w)$ of the same length where the $i$th letter tells
whether the $i$th conjugate $w' = vu$ of $w =uv$ is bordered or not.
Let $[{u}]$ denote the set of conjugates of the word $w$.  We
show that for a 3-letter alphabet $A$, the set of $\beta$-images
equals $\beta(A^n) = B^* \setminus \left([{ab^{n-1}}] \cup
D\right)$ where $D=\{a^n\}$ if $n \in \{5,7,9,10,14,17\}$, and
otherwise $D=\emptyset$.  Hence the number of $\beta$-images is
$B^n_3=2^n-n-m$, where $m=1$ if $n\in \{5,7,9,10,14,17\}$ and $m=0$
otherwise.



\bye

.