On a Generalization of an Inequality of Hardy, Littlewood, and Polya1

These numbers will be regarded as distinct although two or more may be equal in value. Let «,■ (t=l, 2, • • • , k) be any fixed set of k positive integers whose sum is n, let C= Yli-i w»' and let R = n\/C. We shall consider partitions P* = {Bf, B2*, ■ ■ ■ , Bk*\ of the set B into disjoint subsets Bf, each B* containing n,elements of B (* = 1, 2, • • • , k). By a partition here is meant an ordered A-tuplet of subsets, i.e. we consider two partitions P*, P** to be the same (or equal) if and only if B* = B** (» = 1, 2, ■ • •, k). The number of such partitions is given by the multinomial coefficient R. If we let B\ dedenote the first wi elements of B, B\ the next n2 elements of B, • ■ ■ , B\ the last nh elements of B, then Pl= {B\, B\, • • ■ , B\\ is a particular partition of B. Similarly if we let Bf denote the last «i elements of B, B2 the next n2 elements of B, ■ ■ ■ , Bf the first w& elements of B, then PR= {Bf, B2, ■ ■ ■ , Bf} is another particular partition of B. Let A\=Ai (» = 1, 2, • • • , k) be defined similarly for the set A, except that we shall regard the Ai as ordered subsets of A, the order being that given in (1). For convenience we define N= {1, 2, • • • , n) and the subsets Nj=Ni (i=l, 2, • • • , k) exactly as was done for B. The theorem that follows is concerned with the w! cross products XXi ai°ii where iji,j2, ■ • • , jn) is a rearrangement of (1, 2, • ■ • , n). Corresponding to any fixed partition Pr (r = 1, 2, • • • , R) we consider the set VT= {vcr\c = l, 2, • • • , C} of the C cross products obtained by associating the elements of B\ with Ai (i= 1, 2, • • • , k), all possible rearrangements within the subsets B\ being allowed. The n\ cross products are thus divided in R sets with C cross products in each set. Let Pr (r = l, 2, • • ■ , R) denote an arbitrary enumeration of the R partitions except that P1 and PR are defined above. We shall introduce (see The partial ordering below) a partial ordering (written p*>p**) among the partitions such that