Asymptotics and inequalities for the distinct partition function

TL;DR

This paper provides explicit error bounds for the asymptotic expansion of the shifted distinct partition function and uses these to determine exact thresholds for various inequalities related to partition invariants.

Contribution

It introduces refined asymptotic formulas with explicit error bounds for the shifted distinct partition function and applies them to establish precise inequality thresholds.

Findings

Explicit error bounds for the asymptotic expansion of q(n+s)

Exact thresholds for inequalities related to partition invariants

Refined asymptotic formulas for shifted distinct partition functions

Abstract

In this paper, we give explicit error bounds for the asymptotic expansion of the shifted distinct partition function $q(n +s)$ for any nonnegative integer $s$. Then based on this refined asymptotic formula, we give the exact thresholds of $n$ for the inequalities derived from the invariants of the quartic binary form, the double Tur\'{a}n inequalities, the Laguerre inequalities and their corresponding companion versions.