REFINEMENTS OF SOME PARTITION INEQUALITIES

摘要

In the present paper we initiate the study of a certain kind of partition inequality, by showing, for example, that if M 5 is an integer and the integers a and b are relatively prime to M and satisfy 1 ? a < b < M/2, and the c(m, n) are defined by 1 (sqa, sqMa; qM)1 1 (sqb, sqMb; qM)1 := X m,n0 c(m, n)smqn, then c(m,Mn) 0 for all integers m 0, n 0. A similar result is proved for the integers d(m, n) defined by (sqa, sqMa; qM)1 (sqb , sqMb ; qM)1 := X m,n0 d(m, n)smqn. In each case there are obvious interpretations in terms of integer partitions. For example, if p1,5(m, n) (respectively p2,5(m, n)) denotes the number of partitions of n into exactly m parts ±1(mod 5) (respectively ±2(mod 5)), then for each integer n 1, p1,5(m, 5n) p2,5(m, 5n), 1 ? m ? 5n.