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 $$D_{r,t}(n)\sim \frac{3^{\frac{1}{4}}{e}^{\pi \sqrt{\frac{n}{3}}}}{2\pi t{n}^{\frac{1}{4}}}\left(\log (2)+\left(\frac{\sqrt{3}\log (2)}{8\pi }-\frac{\pi }{4\sqrt{3}}\left(r-\frac{t}{2}\right)\right)n^{-\frac{1}{2}}\right)$$ as $n\to \infty$. 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 $n>8$.

arXiv commentCorrects incorrect calculations in Proposition 3.3. Now includes Proposition 3.4