Unimodality of the Rank on Strongly Unimodal Sequences

Let $\{a_i{\}}_{i=1}^{\mathrm{\ell }}$ be a strongly unimodal positive integer sequence with peak position $k$. The rank of such sequence is defined to be $\mathrm{\ell }-2k+1$. Let $u(m,n)$ denote the number of sequences $\{a_i{\}}_{i=1}^{\mathrm{\ell }}$ with rank $m$ and ${\sum }_{i=1}^{\mathrm{\ell }}{a}_i=n$. 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 $n\ge \max \{6,\left(\genfrac{}{}{0px}{}{m+2}{2}\right)\}$. 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 $\mathrm{o}\mathrm{s}\mathrm{p}\mathrm{t}(n)$. Furthermore, using this new combinatorial interpretation, a lower bound and an asymptotic formula on $\mathrm{o}\mathrm{s}\mathrm{p}\mathrm{t}(n)$ will be presented.