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\begin{document}

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\begin{center}
\vskip 1cm{\LARGE\bf
Some Elementary Congruences for  \\
\vskip .05in
the Number of Weighted Integer \\
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Compositions  
}
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\large
Steffen Eger \\
Computer Science Department \\
Goethe University Frankfurt am Main\\
60325 Frankfurt am Main\\
Germany \\
\href{mailto:eger.steffen@gmail.com}{\tt eger.steffen@gmail.com} \\
\ \\
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\begin{abstract}
An integer composition of a nonnegative integer $n$ is a tuple
$(\pi_1,\ldots,\pi_k)$ of nonnegative integers whose sum is $n$; the
$\pi_i$'s are 
called the \emph{parts} of the
composition. For fixed number $k$ of parts, the number of
\emph{$f$-weighted} 
integer compositions (also called \emph{$f$-colored integer
  compositions} in
the literature), in which each part size $s$ may occur in $f(s)$
different colors, is given by the \emph{extended binomial coefficient}
$\binom{k}{n}_{f}$. We derive several congruence properties for
$\binom{k}{n}_{f}$, most of which are analogous to those for ordinary
binomial coefficients. Among them is the parity of $\binom{k}{n}_{f}$,
Babbage's congruence, Lucas' theorem, etc. We also give congruences
for $c_{f}(n)$, the number of $f$-weighted integer compositions with
arbitrarily many parts, and for extended binomial coefficient sums.  
We close with an application of our results to prime criteria for
weighted integer compositions. 
\end{abstract}


\end{document}

                                                                                

