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arXiv · 0805.1622

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

William Y. C. Chen·David G. L. Wang·Iris F. Zhang
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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 $(x, x+m, x+2m, ..., x+(i-1)m) \pmod n$. 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.

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William Y. C. Chen, David G. L. Wang, Iris F. Zhang. 2008-05-12. Partitions of $\mathbb{Z}_n$ into Arithmetic Progressions. https://arxiv.org/abs/0805.1622

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