\begin{abstract}
The method of types is one of the most popular techniques in information theory
and combinatorics. Two sequences of equal length have the same type if
they have identical empirical distributions. In this paper, we focus on
Markov types, that is, sequences generated by a Markov source (of order one).
We note that sequences having the same Markov type
share the same so called {\it balanced frequency matrix} that counts the
number of distinct pairs of symbols. We enumerate the number of
Markov types for sequences of length $n$ over an alphabet of size $m$.
This turns out to coincide with the number of the
balanced frequency matrices as well as with the number of special
{\it linear diophantine equations}, and also balanced directed multigraphs.
For fixed $m$ we prove that the number of Markov types is asymptotically
equal to
$$
d(m)\frac{n^{m^2-m}}{(m^2-m)!},
$$
where $d(m)$ is a constant for which we give an integral representation.
For  $m\to \infty$ we conclude that asymptotically the
number of types is equivalent to
$$
\frac{\sqrt{2}m^{3m/2} e^{m^2}}{m^{2m^2} 2^m \pi^{m/2}} n^{m^2-m}
$$
provided that $m=o(n^{1/4})$ (however, our techniques work for
$m=o(\sqrt{n})$).
We also extend these results to $r$ order Markov sources.
These findings are derived by
analytical techniques ranging from
multidimensional generating functions to the saddle point method.
\end{abstract}