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 in 1972 and submitted it to the "Problems
and Solutions" section of the American Mathematical Monthly, where was
rejected with the terse comment "A bit on the easy side, and using only a standard
argument," presumably because the editors did not understand the actual statement
and solution of the problem (Stanley 2004). The result was therefore first published as Problem 3.75 in
Cohen (1978) after Cohen learned of the result from Stanley. 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.