Asymptotics, Turán inequalities, and the distribution of the BG-rank and 2-quotient rank of partitions

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 $a(\mathrm{mod}b)$. 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 Turán inequalities.

9 pages, with undergrad research supported by the Manitoba eXperimental Mathematics Laboratory. Comments welcome. To appear in Annals of Combinatorics