A perfect partition is a partition of a number whose elements uniquely generate
any number 1, 2, ..., .
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 power, 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).
whose elements uniquely generate
any number 1, 2, ..., 
.
is always a perfect
partition of 
,
and every perfect partition must contain a 1.
.
, 
, 
, 
of 
for 
, 2, ... are given by 1, 1, 2, 1, 3, 1, 4, 2, 3, ... (OEIS
A002033). For 
a prime power, the number
of perfect partitions 
is given by
of 
is equal to the number of ordered
factorizations 
of 
(Goulden and Jackson 1983, p. 94).
