In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic
s
r
a
n
k
(
π
)
=
O
(
π
)
−
O
(
π
′
)
,
\begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*}
where
O
(
π
)
{\mathcal O}(\pi )
denotes the number of odd parts of the partition
π
\pi
and
π
′
\pi ’
is the conjugate of
π
\pi
. In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod
5
5
:
p
0
(
5
n
+
4
)
a
m
p
;
≡
p
2
(
5
n
+
4
)
≡
0
(
mod
5
)
,
p
(
n
)
a
m
p
;
=
p
0
(
n
)
+
p
2
(
n
)
,
\begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*}
where
p
i
(
n
)
p_i(n)
(
i
=
0
,
2
i=0,2
) denotes the number of partitions of
n
n
with
s
r
a
n
k
≡
i
(
mod
4
)
\mathrm {srank}\equiv i\pmod {4}
and
p
(
n
)
p(n)
is the number of unrestricted partitions of
n
n
. Andrews asked for a partition statistic that would divide the partitions enumerated by
p
i
(
5
n
+
4
)
p_i(5n+4)
(
i
=
0
,
2
i=0,2
) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the
2
2
-quotient-rank and the
5
5
-core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the
2
2
-quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod
5
5
. This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo
5
5
. Finally, we discuss some new formulas for partitions that are
5
5
-cores and discuss an intriguing relation between
3
3
-cores and the Andrews-Garvan crank.