Partitions of ℤ_n into Arithmetic Progressions

2008/05/12 by William Y. C. Chen, David G. L. Wang, Chen, William Y. C. +3
Mathematics · #05A05 #05A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A05 #msc:05A15

paper · pdf · doi:10.48550/arxiv.0805.1622

11 pages, 2 figures

arxiv created 2008/05/12 · openalex publication_date 2008/05/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce the notion of arithmetic progression blocks or AP-blocks of 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 ℤ_n into AP-blocks for a given difference m. We show that subject to a technical condition, the number of partitions of 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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