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, sqM−a; qM)∞ − 1 (sqb , sqM−b ; qM)∞ := X m,n≥0 c(m, n)s mq n , 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 , −sqM−a ; q M)∞ − (−sqb , −sqM−b ; q M)∞ := X m,n≥0 d(m, n)s mq n . 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.
INTEGERS: The Electronic Journal of Combinatorial Number Theory
Colgate University, Charles University, and DIMATIA
McLaughlin, J. (2016). Refinements of Some Partition Inequalities. INTEGERS: The Electronic Journal of Combinatorial Number Theory, 16(A66), 1-16. Retrieved from https://digitalcommons.wcupa.edu/math_facpub/70