On the Number of Parts in Congruence Classes for Partitions into Distinct Parts

TL;DR

This paper derives an asymptotic formula for the number of parts in partitions into distinct parts congruent to a residue modulo t, and establishes inequalities between these counts for large n, with explicit bounds for small t.

Contribution

It provides the first asymptotic formula for D_{r,t}(n) and proves inequalities between counts for different residues, including effective bounds for small t.

Findings

Asymptotic formula for D_{r,t}(n) as n grows large.

Inequality D_{r,t}(n) ≥ D_{s,t}(n) holds for large n when 0<r<s≤t.

Explicit bounds show the inequality holds for all n > 8 when 2 ≤ t ≤ 10.

Abstract

For integers $0 < r \leq t$, 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 $$D_{r,t}(n) \sim \dfrac{3^{\frac 14} e^{\pi \sqrt{\frac{n}{3}}}}{2\pi t n^{\frac 14}} \left( \log(2) + \left( \dfrac{\sqrt{3} \log(2)}{8\pi} - \dfrac{\pi}{4\sqrt{3}} \left( r - \dfrac{t}{2} \right) \right) n^{- \frac 12} \right)$$ as $n \to \infty$. A corollary of this result is that for $0 < r < s \leq t$, the inequality $D_{r,t}(n) \geq D_{s,t}(n)$ holds for all sufficiently large $n$. We make this effective, showing that for $2 \leq t \leq 10$ the inequality $D_{r,t}(n) \geq D_{s,t}(n)$ holds for all $n > 8$.