Partitions of into arithmetic progressions

We introduce the notion of arithmetic progression blocks or m -AP-blocks of Z n , which can be represented as sequences of the form ( x , x + m , x + 2 m , … , x + ( i − 1 ) m ) ( mod n ) . Then we consider the problem of partitioning Z n into m -AP-blocks. 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 , 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 Z n 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.