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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 subjectto 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 ourattention to blocks of sizes one or two, we are led to a combinatorialinterpretation of a formula recently derived by Mansour and Sun as ageneralization of the Kaplansky numbers. These numbers have also occurred asthe coefficients in Waring's formula for symmetric functions.

Partitions of $\mathbb{Z}_n$ into Arithmetic Progressions