Abstract
Let
S
(
n
)
$S(n)$
upper S left parenthesis n right parenthesis
be the number of partitions
a
1
+
a
2
+
a
3
+
⋯
$a_1+a_2+a_3+\cdots $
a 1 plus a 2 plus a 3 plus midline horizontal ellipsis
with
a
1
≥
a
2
≥
a
3
≥
⋯
$a_1 \geq a_2 \geq a_3 \geq \cdots $
a 1 greater than or equals a 2 greater than or equals a 3 greater than or equals midline horizontal ellipsis
such that
n
=
a
1
+
a
3
+
a
5
+
⋯
$n = a_1+a_3+a_5+\cdots $
n equals a 1 plus a 3 plus a 5 plus midline horizontal ellipsis
and
a
1
,
a
3
,
a
5
,
…
$a_1, a_3, a_5, \ldots $
a 1 comma a 3 comma a 5 comma ellipsis
are all even. We apply the action of Atkin’s
U
7
$U_7$
upper U 7
operator to a certain
η
$\eta $
eta
-quotient for the congruence subgroup
Γ
0
(
98
)
$\Gamma _0(98)$
normal upper Gamma 0 left parenthesis 98 right parenthesis
and a result of Ahlgren to produce infinite families of congruences modulo
7
$7$
7
for
S
(
n
)
$S(n)$
upper S left parenthesis n right parenthesis
.