Asymptotic equidistribution and convexity for partition ranks

We study the partition rank function N(r, t; n), the number of partitions with rank congruent to r modulo t. We first show that it is monotonic in n above a given bound, and then show that it equidistributed as $$n \rightarrow \infty $$. Using this result we prove a conjecture of Hou and Jagadeeson on the convexity of N(r, t; n).