On the number of parts in congruence classes for partitions into distinct parts

For integers $$0 < r \le t$$ , let the function $$D_{r,t}(n)$$ denote the number of parts among all partitions of n into distinct parts that are congruent to r modulo t. We prove the asymptotic formula $$\begin{aligned} D_{r,t}(n) \sim \dfrac{3^{\frac{1}{4}} e^{\pi \sqrt{\frac{n}{3}}}}{2\pi t n^{\frac{1}{4}}} \left( \log (2) + \left( \dfrac{\sqrt{3} \log (2)}{8\pi } - \dfrac{\pi }{4\sqrt{3}} \left( r - \dfrac{t}{2} \right) \right) n^{- \frac{1}{2}} \right) \end{aligned}$$ as $$n \rightarrow \infty $$ . A corollary of this result is that for $$0< r < s \le t$$ , the inequality $$D_{r,t}(n) \ge D_{s,t}(n)$$ holds for all sufficiently large n. We make this effective, showing that for $$2 \le t \le 10$$ the inequality $$D_{r,t}(n) \ge D_{s,t}(n)$$ holds for all $$n > 8$$ .