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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 parts of (i.e., numbers in) those Partitions.
See also Elder's Theorem, Partition
References
Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer, pp. 6-8, 1985.