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
.