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# Perfect Partition

A perfect partition is a partition of a number whose elements uniquely generate any number 1, 2, ..., . is always a perfect partition of , and every perfect partition must contain a 1.

The following table gives the first several perfect partitions for small .

 perfect partitions 1 1 2 1 3 2 , 4 1 5 3 , , 6 1

The numbers of perfect partitions of for , 2, ... are given by 1, 1, 2, 1, 3, 1, 4, 2, 3, ... (OEIS A002033). For a prime power, the number of perfect partitions is given by

The number of perfect partitions of is equal to the number of ordered factorizations of (Goulden and Jackson 1983, p. 94).

Ordered Factorization, Partition

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## References

Cohen, D. I. A. Basic Techniques of Combinatorial Theory. New York: Wiley and Sons, p. 97, 1978.Goulden, I. P. and Jackson, D. M. Problem 2.5.12 in Combinatorial Enumeration. New York: Wiley, 1983.Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 140-143, 1985.Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1958.Sloane, N. J. A. Sequences A002033/M0131 and A035341 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Perfect Partition

## Cite this as:

Weisstein, Eric W. "Perfect Partition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PerfectPartition.html