Dyson's ranks and Maass forms

Motivated by work of Ramanujan, Freeman Dyson defined the rank of an integer partition to be its largest part minus its number of parts. If N(m, n) denotes the number of partitions of n with rank m, then it turns out that R(w; q) : = 1 + Sigma(infinity)(n=1) Sigma(infinity)(m=-infinity) N(m, n)w(m)q(n) = 1 + Sigma(infinity)(n=1) q(n2)/Pi(n)(j=1)(1-(w+w(-1))q(j) + q(2j)). We show that if zeta not equal 1 is a root of unity, then R(zeta; q) is essentially the holomorphic part of a weight 1/2 weak Maass form on a subgroup of SL2(Z). For integers 0 <= r < t, we use this result to determine the modularity of the generating function for N(r, t; n), the number of partitions of n whose rank is congruent to r (mod t). We extend the modularity above to construct an infinite family of vector valued weight 1/2 forms for the full modular group SL2(Z), a result which is of independent interest.