  A variety of associative algebras over a field of characteristic 0 is called
minimal if its codimension sequence grows much faster than the codimension
sequence of any of its proper subvarieties. By the results of Giambruno and
Zaicev it follows that the number $b_n$ of minimal varieties of given exponent
$n$ is finite. Using methods of the theory of colored (or weighted)
compositions of integers, we show that the limit
$\beta=\lim_{n\to\infty}\sqrt[n]{b_n}$ exists and can be expressed as the
positive solution of an equation $a(t)=0$ where $a(t)$ is an explicitly given
power series. Similar results are obtained for the number of minimal varieties
with a given Gelfand-Kirillov dimension of their relatively free algebras of
rank $d$. It follows from classical results on lacunary power series that the
generating function of the sequence $b_n$, $n=1,2,\ldots$, is transcendental.
With the same approach we construct examples of free graded semigroups $\langle
Y\rangle$ with the following property. If $d_n$ is the number of elements of
degree $n$ of $\langle Y\rangle$, then the limit
$\delta=\lim_{n\to\infty}\sqrt[n]{d_n}$ exists and is transcendental.
