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A generalization of Stanley's Theorem. It 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.
See also Stanley's Theorem
References
Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer, pp. 8-9, 1985.