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

In this paper, we give an iterated approach to concern with the positivity of det ( p ( n − i + j ) ) 1 ≤ i , j ≤ k , \begin{equation*} \det \ (p(n-i+j))_{1\leq i,j\leq k}, \end{equation*} where p ( n ) 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 k_1,k_2,m_1,m_2 , one can find a threshold N ( k 1 , k 2 , m 1 , m 2 ) N(k_1,k_2,m_1,m_2) such that for n > N ( k 1 , k 2 , m 1 , m 2 ) n>N(k_1,k_2,m_1,m_2) , | p ( n − k 1 + m 1 ) a m p ; p ( n + m 1 ) a m p ; p ( n + m 1 + m 2 ) p ( n − k 1 ) a m p ; p ( n ) a m p ; p ( n + m 2 ) p ( n − k 1 − k 2 ) a m p ; p ( n − k 2 ) a m p ; p ( n − k 2 + m 2 ) | > 0. \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 ≥ 656 n\geq 656 , det ( p ( n − i + j ) ) 1 ≤ i , j ≤ 4 > 0 \det \ (p(n-i+j))_{1\leq i,j\leq 4}>0 . Employing the same technique, we will show that determinants ( p ¯ ( n − i + j ) ) 1 ≤ i , j ≤ k ({\bar p}(n-i+j))_{1\leq i,j\leq k} are positive for k = 3 and 4 k=3 \text { and } 4 for overpartition p ¯ ( n ) {\bar p}(n) . Furthermore, we will give an outline of how to prove the positivity of det ( p ( n − i + j ) ) 1 ≤ i , j ≤ k \det \ (p(n-i+j))_{1\leq i,j\leq k} for general k k .