Positivity of the determinants of the partition function and the overpartition function

Abstract:

In this paper, we give an iterated approach to concern with the positivity of \begin{equation*} \det \ (p(n-i+j))_{1\leq i,j\leq k}, \end{equation*} where $p(n)$ is the partition function. We first apply a general method to prove that for given $k_1,k_2,m_1,m_2$, one can find a threshold $N(k_1,k_2,m_1,m_2)$ such that for $n>N(k_1,k_2,m_1,m_2)$, \begin{equation*} \begin {vmatrix} p(n-k_1+m_1) & p(n+m_1) & p(n+m_1+m_2)\\ p(n-k_1) & p(n) & p(n+m_2)\\ p(n-k_1-k_2) & p(n-k_2) & p(n-k_2+m_2) \end{vmatrix}>0. \end{equation*} Based on this result, we will prove that for $n\geq 656$, $\det \ (p(n-i+j))_{1\leq i,j\leq 4}>0$. Employing the same technique, we will show that determinants $({\bar p}(n-i+j))_{1\leq i,j\leq k}$ are positive for $k=3 \text { and } 4$ for overpartition ${\bar p}(n)$. Furthermore, we will give an outline of how to prove the positivity of $\det \ (p(n-i+j))_{1\leq i,j\leq k}$ for general $k$.