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

Abstract

We introduce the notion of arithmetic progression blocks or AP-blocks of ${\mathbb{Z}}_n$, which can be represented as sequences of the form $(x,x+m,x+2m,...,x+(i-1)m)(\mathrm{mod}n)$. 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