Asymptotics, Tur\'an inequalities, and the distribution of the BG-rank and 2-quotient rank of partitions

TL;DR

This paper investigates the asymptotic behavior and inequalities related to the distribution of BG-rank and 2-quotient rank in partitions, revealing equidistribution and Turán inequalities for these statistics.

Contribution

It provides the first asymptotic formulas for BG-rank and 2-quotient rank distributions and proves they satisfy higher-order Turán inequalities.

Findings

Asymptotic formulas for BG-rank and 2-quotient rank distributions.

Proof of asymptotic equidistribution over congruence classes.

Validation that these distributions satisfy higher-order Turán inequalities.

Abstract

Let $j,n$ be even positive integers, and let $\overline{p}_j(n)$ denote the number of partitions with BG-rank $j$, and $\overline{p}_j(a,b;n)$ to be the number of partitions with BG-rank $j$ and $2$-quotient rank congruent to $a \pmod{b}$. We give asymptotics for both statistics, and show that $\overline{p}_j(a,b;n)$ is asymptotically equidistributed over the congruence classes modulo $b$. We also show that each of $\overline{p}_j(n)$ and $\overline{p}_j(a,b;n)$ asymptotically satisfy all higher-order Tur\'{a}n inequalities.