In this paper, we explore intricate connections between Ramanujan’s theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation with the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.
Let p(n) denote the number of partitions of n. A new infinite family of inequalities for p(n) is presented. This generalizes a result by William Chen et al. From this infinite family, another infinite family of inequalities for $$\log p(n)$$
log
p
(
n
)
is derived. As an application of the latter family one, for instance obtains that for $$n\ge 120$$
n
≥
120
, $$\begin{aligned} p(n)^2>\Biggl (1+\frac{\pi }{\sqrt{24}n^{3/2}}-\frac{1}{n^2}\Biggr )p(n-1)p(n+1). \end{aligned}$$
p
(
n
)
2
>
(
1
+
π
24
n
3
/
2
-
1
n
2
)
p
(
n
-
1
)
p
(
n
+
1
)
.
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