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

Chen, William Y. C.; Wang, David G. L.; Zhang, Iris F.

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

Authors

Chen, William Y. C.

Wang, David G. L.

Zhang, Iris F.

Description

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.
11 pages, 2 figures