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

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