Abstract
Let
$\mathcal {O}(\pi )$
denote the number of odd parts in an integer partition
$\pi$
. In 2005, Stanley introduced a new statistic
$\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$
, where
$\pi '$
is the conjugate of
$\pi$
. Let
$p(r,\,m;n)$
denote the number of partitions of
$n$
with srank congruent to
$r$
modulo
$m$
. Generating function identities, congruences and inequalities for
$p(0,\,4;n)$
and
$p(2,\,4;n)$
were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for
$p(r,\,m;n)$
with
$m=16$
and
$24$
. These results are refinements of some inequalities due to Swisher.
On the Growth of Restricted Integer Partition Functions