Partitions of ZnZn into arithmetic progressions

We introduce the notion of arithmetic progression blocks or mm-AP-blocks of ZnZn, which can be represented as sequences of the form (x,x+m,x+2m,…,x+(i−1)m)(modn). Then we consider the problem of partitioning ZnZn into mm-AP-blocks. We show that subject to a technical condition, the number of partitions of ZnZn into mm-AP-blocks of a given type is independent of mm, 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 ZnZn 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+1p+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.