Inequalities for ranks of partitions and the first moment of ranks and cranks of partitions

摘要

We prove two monotonicity properties of N(m, n), the number of partitions of n with rank m. They are (i) for any nonnega- tive integers m and n, N(m, n) ≥ N(m + 2,n), and, (ii) for any nonnegative integers m and n such that n ≥ 12, n ≠ m + 2, N (m, n) ≥ N(m, n ? 1). G.E. Andrews, B. Kim, and the first author introduced ospt(n), a function counting the difference between the first positive rank and crank moments. They proved that ospt(n) > 0. In another article, K. Bringmann and K. Mahl- burg gave an asymptotic estimate for ospt(n). The two mono- tonicity properties for N(m,n) lead to stronger inequalities for ospt(n) that imply the asymptotic estimate.