On arithmetic partitions of Z_n

TL;DR

This paper improves existing theorems on partitioning cyclic groups into arithmetic progressions by applying a convolution formula for cyclic multinomial coefficients, advancing combinatorial enumeration techniques.

Contribution

It introduces a new approach using Raney-Mohanty's convolution formula to strengthen results on arithmetic partitions of cyclic groups.

Findings

Enhanced enumeration formulas for arithmetic partitions of Z_n.

Application of convolution formula to improve previous theorems.

Broader understanding of combinatorial structures in cyclic groups.

Abstract

Generalizing a classical problem in enumerative combinatorics, Mansour and Sun counted the number of subsets of $\Z_n$ without certain separations. Chen, Wang, and Zhang then studied the problem of partitioning $\Z_n$ into arithmetical progressions of a given type under some technical conditions. In this paper, we improve on their main theorems by applying a convolution formula for cyclic multinomial coefficients due to Raney-Mohanty.