  We introduce the notion of arithmetic progression blocks or AP-blocks of
$\mathbb{Z}_n$, which can be represented as sequences of the form $(x, x+m,
x+2m, ..., x+(i-1)m) \pmod n$. Then we consider the problem of partitioning
$\mathbb{Z}_n$ into AP-blocks for a given difference $m$. We show that subject
to a technical condition, the number of partitions of $\mathbb{Z}_n$ into
$m$-AP-blocks of a given type is independent of $m$. When we restrict our
attention to blocks of sizes one or two, we are led to a combinatorial
interpretation of a formula recently derived by Mansour and Sun as a
generalization of the Kaplansky numbers. These numbers have also occurred as
the coefficients in Waring's formula for symmetric functions.
