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:

Cohen, D. I. A. Basic Techniques of Combinatorial Theory. New York: Wiley and Sons, 1978.Hoare,
A. H. M. "An Involution of Blocks in the Partitions of ." Amer. Math. Monthly93, 475-476, 1986.Honsberger,
R. Mathematical
Gems III. Washington, DC: Math. Assoc. Amer, pp. 8-9, 1985.Kirdar,
M. S. and Skyrme, T. H. R. "On an Identity Related to Partitions
and Repetitions of Parts." Canad. J. Math.34, 194-195, 1982.Sloane,
N. J. A. Sequence A066633 in "The
On-Line Encyclopedia of Integer Sequences."Stanley, R. P.
Exercise 1.26 in Enumerative
Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press,
p. 59, 1999.Stanley, R. P. "Errata and Addenda to Enumerative
Combinatorics Volume 1, Second Printing." Rev. Feb. 13, 2004. http://www-math.mit.edu/~rstan/ec/newerr.ps.