Restricted Partitions of Finite Sets

Abstract

In this paper we consider the following combinatorial problem. In how many ways can n distinguishable objects be placed into an unrestricted number of indistinguishable boxes, if each box can hold at most r objects? Let us denote this number by G n, r Special cases of this problem have been the object of considerable study. In the case r = 2 we have the numbers G n, 2 = T n which have been treated by Rothe [12] as early as 1800. T n is also the number of solutions of x 2 = 1 in the symmetric group on n letters , and in this and related guises has been studied by Touchard [13], Chowla, Herstein and Moore [3] and two of the present authors [7].