The number of partitions of 
in which no parts are multiples of 
is sometimes denoted 
(Gordon and Ono 1997). 
is also the number of partitions of 
into at most 
copies of each part.
There is a special case
 |
(1)
|
where 
is the partition function Q, and 
is the number of irreducible 
-modular representations of the symmetric
group 
.
The generating function for 
is given by
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is a q-Pochhammer symbol.
The following table gives the first few values of 
for small 
.
 | OEIS |  |
| 2 | A000009 | 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, ... |
| 3 | A000726 | 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 57, ... |
| 4 | A001935 | 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, ... |
| 5 | A035959 | 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, ... |
Gordon and Ono (1997) show that
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
Defining 
as the number of positive integers 
for which 
, Gordon and Ono (1997) proved that if 
, then
 |
(7)
|
for all 
,
where 
.
