Inequalities for Differences of Dyson’s Rank for All Odd Moduli

A partition of a non-negative integer n is any non-increasing sequence of positive integers whose sum is n. As usual, let p(n) denote the number of partitions of n. The partition function satisfies the famous “Ramanujan congruences” declaring that for c ∈ {5, 7, 11} we have for all n ≥ 0 that p(cn+δc) ≡ 0 (mod c), where δc is defined by the congruence 24δc ≡ 1 (mod c). In order to understand these from a combinatorial point of view, Dyson defined the rank of a partition as its largest part minus its number of parts [11]. Atkin and Swinnerton-Dyer [4] later proved that Dyson’s rank indeed provides a combinatorial explanation of the congruences modulo 5 and 7, but not the congruence modulo 11. To simplify notation, for integers 0 ≤ a < c, we let N(a, c;n) to be the number of partitions of n whose rank is congruent to a (mod c). Rank differences have been the focus of several works and lead to interesting new automorphic forms, so called harmonic Maass forms. Harmonic Maass forms are generalizations of modular forms, in that they satisfy the same transformation law, and (weak) growth conditions at cusps, but instead of being holomorphic, they are annihilated by the weight k hyperbolic Laplacian. As an example, consider the function