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

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

2008-05-12

arxiv.org/abs/0805.1622v1

Mathematics

Combinatorics

11 pages, 2 figures

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Partitions of $\mathbb{Z}_n$ into Arithmetic Progressions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Partitions of $\mathbb{Z}_n$ into Arithmetic Progressions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Partitions of $\mathbb{Z}_n$ into Arithmetic Progressions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-325236