Partitions of Zn into Arithmetic Progressions
William Y.C. Chen, David G.L. Wang and Iris F. Zhang
Abstract: We introduce the notion of arithmetic progression blocks or m-AP-blocks of Zn, which can be represented as sequences of the form (x, x+m, x+2m,..., x+ (i-1)m) (mod n). Then we consider the problem of partitioning Zn into m-AP-blocks. We show that subject to a technical condition, the number of partitions of Zn into m-AP-blocks of a given type is independent of m, and is equal to the cyclic multinomial coefficient which has occurred in Waring's formula for symmetric functions. The type of such a partition of Zn is defined by the type of the underlying set partition. We give a combinatorial proof of this formula and the construction is called the separation algorithm. When we restrict our attention to blocks of sizes 1 and p+1, we are led to a combinatorial interpretation of a formula recently derived by Mansour and Sun as a generalization of the Kaplansky numbers. By using a variant of the cycle lemma, we extend the bijection to deal with an improvement of the technical condition recently given by Guo and Zeng.
AMS Classification: 05A05, 05A15, 11B50
Keywords: Kaplansky number, cycle dissection, m-AP-partition, separation algorithm