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

Abstract

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 $n > 8$.