  A vertex coloring of a graph $G$ is called distinguishing (or symmetry
breaking) if no non-identity automorphism of $G$ preserves it, and the
distinguishing number, shown by $D(G)$, is the smallest number of colors
required for such a coloring. This paper is about counting non-equivalent
distinguishing colorings of graphs with $k$ colors. A parameter, namely $\Phi_k
(G)$, which is the number of non-equivalent distinguishing colorings of a graph
$G$ with at most $k$ colors, is shown here to have an application in
calculating the distinguishing number of the lexicographic product and the
$X$-join of graphs. We study this index (and some other similar indices) which
is generally difficult to calculate. Then, we show that if one knows the
distinguishing threshold of a graph $G$, which is the smallest number of colors
$\theta(G)$ so that, for $k\geq \theta(G)$, every $k$-coloring of $G$ is
distinguishing, then, in some special cases, counting the number of
distinguishing colorings with $k$ colors is very easy. We calculate $\theta(G)$
for some classes of graphs including the Kneser graph $K(n,2)$. We then turn to
vertex partitioning by studying the distinguishing coloring partition of a
graph $G$; a partition of vertices of $G$ which induces a distinguishing
coloring for $G$. There, we introduce $\Psi_k (G)$ as the number of
non-equivalent distinguishing coloring partitions with at most $k$ cells, which
is a generalization to its distinguishing coloring counterpart.
