Unimodality of the Rank on Strongly Unimodal Sequences

Let $\{a_i\}_{i=1}^\ell$ be a strongly unimodal positive integer sequence with peak position $k$. The rank of such sequence is defined to be $\ell-2k+1$. Let $u(m,n)$ denote the number of sequences $\{a_i\}_{i=1}^\ell$ with rank $m$ and $\sum_{i=1}^{\ell} a_i=n$. Bringmann, Jennings-Shaffer, Mahlburg and Rhoades conjectured that $\{u(m,n)\}_m$ is strongly log-concave for any fixed $n$. Motivated by this conjecture, in this paper we prove the strongly unimodality of $\{u(m,n)\}_m$, that is $u(m,n)>u(m+1,n)$ for $m\ge 0$ and $n\ge \max\{6,{m+2\choose 2}\}$. This result gives supportive evidence for the above conjecture. Moreover, we find a combinatorial interpretation of $u(m,n)$, which leads to a new combinatorial interpretation of ${\rm ospt}(n)$. Furthermore, using this new combinatorial interpretation, a lower bound and an asymptotic formula on ${\rm ospt}(n)$ will be presented.