Let
p
(
n
1
,
…
,
n
j
:
A
1
,
…
,
A
j
)
p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})
be the number of partitions of
(
n
1
,
…
,
n
j
)
({n_1}, \ldots ,{n_j})
where, for
1
⩽
l
⩽
j
1 \leqslant l \leqslant j
, the lth component of each part belongs to the set
A
l
=
⋃
h
(
l
)
=
1
q
(
l
)
{
a
l
h
(
l
)
+
M
v
:
v
=
0
,
1
,
2
,
…
}
{A_l} = \bigcup \nolimits _{h(l) = 1}^{q(l)} {\{ {a_{lh(l)}} + Mv :v = 0,1,2, \ldots \} }
and
M
,
q
(
l
)
M,q(l)
and the
a
l
h
(
l
)
{a_{lh(l)}}
are positive integers such that
0
>
a
l
1
>
⋯
>
a
l
q
(
l
)
⩽
M
0 > {a_{l1}} > \cdots > {a_{lq(l)}} \leqslant M
. Asymptotic expansions for
p
(
n
1
,
…
,
n
j
:
A
1
,
…
,
A
j
)
p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})
are derived, when the
n
l
→
∞
{n_l} \to \infty
subject to the restriction that
n
1
⋯
n
j
⩽
n
l
j
+
1
−
∈
{n_1} \cdots {n_j} \leqslant n_l^{j + 1 - \in }
for all l, where
∈
\in
is any fixed positive number. The case
M
=
1
M = 1
and arbitrary j was investigated by Robertson [10] while several authors between 1940 and 1960 investigated the case
j
=
1
j = 1
for different values of M.