Proof of two conjectures on the log-concavity of partition ranks and cranks

Abstract

Let $M(m,n)$ and $N(m,n)$ denote the numbers of partitions of $n$ with crank $m$ and rank $m$, respectively. Ji and Zang proved that the crank sequence $(M(m,n))_{|m|\le n-1}$ is unimodal for every $n\ge44$. Bringmann, Man, and Rolen subsequently established the unimodality of the rank sequence $(N(m,n))_{|m|\le n-2}$ for every $n\ge39$. Ji and Zang also conjectured that $(M(m,n))_{|m|\le n-71}$ is log-concave for every $n\ge72$, and that $(N(m,n))_{|m|\le n-72}$ is log-concave for every $n\ge73$. In this paper, we prove both conjectures.