  We determine the number of isomorphism classes of elementary gradings by a
finite group on an algebra of upper block-triangular matrices. As a consequence
we prove that, for a finite abelian group $G$, the sequence of the numbers
$E(G,m)$ of isomorphism classes of elementary $G$-gradings on the algebra
$M_{m}(\mathbb{F})$ of $m\times m$ matrices with entries in a field
$\mathbb{F}$ characterizes $G$. A formula for the number of isomorphism classes
of gradings by a finite abelian group on an algebra of upper block-triangular
matrices over an algebraically closed field, with mild restrictions on its
characteristic, is also provided. Finally, if $G$ is a finite abelian group,
$\mathbb{F}$ is an algebraically closed field and $N(G,m)$ is the number of
isomorphism classes of $G$-gradings on $M_{m}% (\mathbb{F})$ we prove that
$N(G,m)\sim\frac{1}{\left\vert G\right\vert !}m^{\left\vert G\right\vert
-1}\sim E(G,m)$.
