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

TL;DR

This paper studies how to partition the cyclic group bZ_n into arithmetic progression blocks, revealing independence of the partition count from the difference parameter under certain conditions and connecting to combinatorial and algebraic formulas.

Contribution

It introduces AP-blocks of bZ_n, proves the independence of partition counts from the difference parameter, and links these results to known combinatorial and algebraic formulas.

Findings

Partition counts are independent of the difference m under a technical condition.

A combinatorial interpretation of generalized Kaplansky numbers is provided.

Connections to Waring's formula for symmetric functions are established.

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.