A perfect partition is a partition of a positive integer 
whose parts contain exactly one partition of each integer from 1 through 
, with repeated parts regarded as indistinguishable. 
is always a perfect partition of 
, and every perfect partition must contain a 1.
The following table gives the first several perfect partitions for small 
.
 |  | perfect partitions |
| 1 | 1 |  |
| 2 | 1 |  |
| 3 | 2 |  ,  |
| 4 | 1 |  |
| 5 | 3 |  ,  ,  |
| 6 | 1 |  |
The numbers of perfect partitions 
of 
for 
, 2, ... are given by 1, 1, 2, 1, 3, 1, 4, 2, 3, ... (OEIS
A002033). For a prime
number 
and a positive integer 
, the number of perfect partitions 
is given by
 |
The number 
of perfect partitions of 
is equal to the number of ordered
factorizations 
of 
(Goulden and Jackson 1983, p. 94).
