Partitions of $\mathbb{Z}_n$ into Arithmetic Progressions
| dc.creator | Chen, William Y. C. | |
| dc.creator | Wang, David G. L. | |
| dc.creator | Zhang, Iris F. | |
| dc.date | 2008-05-12 | |
| dc.date.accessioned | 2026-07-07T09:38:20Z | |
| dc.date.available | 2026-07-07T09:38:20Z | |
| dc.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. | |
| dc.description | 11 pages, 2 figures | |
| dc.identifier | https://arxiv.org/abs/0805.1622 | |
| dc.identifier | http://arxiv.org/abs/0805.1622 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/160771 | |
| dc.subject | Combinatorics | |
| dc.subject | 05A05, 05A15 | |
| dc.title | Partitions of $\mathbb{Z}_n$ into Arithmetic Progressions | |
| dc.type | text |