  The paper studies the $K$-theoretic invariants of the crossed product
$C^{*}$-algebras associated with an important family of homeomorphisms of the
tori $\Bbb{T}^{n}$ called {\em Furstenberg transformations}. Using the
Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those
crossed products, whose corresponding $n\times n$ integer matrices are
unipotent of maximal degree, always have the same rank $a_{n}$. We show using
the theory developed here, together with two computing programs - included in
an appendix - that a claim made in the literature about the torsion subgroups
of these $K$-groups is false. Using the representation theory of the simple Lie
algebra $\frak{sl}(2,\Bbb{C})$, we show that, remarkably, $a_{n}$ has a
combinatorial significance. For example, every $a_{2n+1}$ is just the number of
ways that 0 can be represented as a sum of integers between $-n$ and $n$ (with
no repetitions). By adapting an argument of van Lint (in which he answered a
question of Erd\"os), a simple, explicit formula for the asymptotic behavior of
the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of
the K_{0}-groups of an important class of Furstenberg crossed products,
obtaining their complete Elliott invariant using classification results of H.
Lin and N. C. Phillips.
