Elder's Theorem

Elder's theorem states that, for positive integers and , the total number of occurrences of the part among all unordered partitions of equals the total number of distinct part sizes that occur at least times. A partition containing distinct part sizes with multiplicity at least contributes to the latter total. The theorem generalizes Stanley's theorem, the case .

Stanley (2021, p. 2) reports that he discovered the general result in 1972 and submitted it to the "Problems and Solutions" section of the American Mathematical Monthly, where it was rejected with the comment "A bit on the easy side, and using only a standard argument." Stanley conjectures that the editors did not understand the actual statement and solution. The result was 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." Independent proofs of the general case were given by Kirdar and Skyrme (1982), Paul Elder in 1984 (as reported by Honsberger 1985, p. 8), and Hoare (1986).

As an example of the theorem, note that the partitions of 4 are 4, , , , and , which contain ones, twos, three, and four. Similarly, a part occurs 1 or more times on occasions, 2 or more times on occasions, 3 or more times on occasion, and 4 or more times on occasion.

In general, the numbers of times that 1, 2, ..., occur in the partitions of are given by the following triangle:

	1	2	3	4	5	6	7	8
1	1
2	2	1
3	4	1	1
4	7	3	1	1
5	12	4	2	1	1
6	19	8	4	2	1	1
7	30	11	6	3	2	1	1
8	45	19	9	6	3	2	1	1

(OEIS A066633).