Unimodality of the Rank on Strongly Unimodal Sequences

Let ᵢ\ᵢ₌₁^ be a strongly unimodal positive integer sequence with peak position k. The rank of such sequence is defined to be -2k+1. Let $u(m,n)$ denote the number of sequences ᵢ\ᵢ₌₁^ with rank m and ∑ᵢ₌₁^ aᵢ=n. Bringmann, Jennings-Shaffer, Mahlburg and Rhoades conjectured that (m,n)\ₘ is strongly log-concave for any fixed n. Motivated by this conjecture, in this paper we prove the strongly unimodality of (m,n)\ₘ, that is $u(m,n)>u(m+1,n)$ for m≥ 0 and n≥ max\6,m+2 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 ospt(n). Furthermore, using this new combinatorial interpretation, a lower bound and an asymptotic formula on ospt(n) will be presented.