Elder's Theorem

Elder's theorem is a generalization of Stanley's theorem which states that the total number of occurrences of an integer among all unordered partitions of is equal to the number of occasions that a part occurs or more times in a partition, where a partition which contains parts that each occur or more times contributes to the sum in question.

The general result 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 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." 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 contains 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).