Stanley's theorem states that the total number of 1s that occur among all unordered partitions of a positive
integer is equal to the sum of the numbers of distinct members of those partitions.
The generalization sometimes known as Elder's theorem
was discovered by R. P. Stanley, who reports that he submitted it to the
"Problems and Solutions" section of the American Mathematical Monthly
in 1972 (Stanley 2021, p. 2). According to Stanley, it was rejected with the
comment "A bit on the easy side, and using only a standard argument." Stanley
speculates that the editors did not understand the actual statement and solution.
After learning of the result from Stanley, Cohen (1978) included the 
case as Problem 3.75. For this reason, the case 
is sometimes called "Stanley's theorem."
As an example of the theorem, note that the partitions of 5 are 
,
, 
, 
, 
, 
, 
. There are a total of 
1s in this list, which is equal to the sums
of the numbers of unique terms in each partition: 
.
The numbers of 1s occurring in all partitions of 
, 2, 3, ... are 1, 2, 4, 7, 12, 19,
30, 45, 67, ... (OEIS A000070). The 
th term of this sequence is also equal to 
, where 
is the partition function
P.
