                                    Partition     A partition is a way of writing an integer n as a sum of positive integers where the order of the addends is not significant, possibly subject to one or more additional constraints. By convention, partitions are normally written from largest to smallest addends (Skiena 1990, p. 51), for example, . All the partitions of a given positive integer n can be generated using Partitions[n] in the Mathematica add-on package DiscreteMath`Combinatorica` (which can be loaded with the command <<DiscreteMath`), or the undocumented Mathematica command DiscreteMath`IntegerPartitions`IntegerPartitions[n]. PartitionsQ[p] can be used to test if a list consists of positive integers and therefore is a valid partition.

Andrews (1998, p. 1) used the notation to indicate "a sequence is a partition of n," and the notation to abbreviate the partition . The partitions on a number n correspond to the set of solutions to the Diophantine equation For example, the partitions of four, given by (1, 1, 1, 1), (1, 1, 2), (2, 2), (4), and (1, 3) correspond to the solutions , (2, 1, 0, 0), (0, 2, 0, 0), (0, 0, 0, 1), and (1, 0, 1, 0).

Particular types of partition functions include the partition function P, giving the number of partitions of a number as a sum of smaller integers without regard to order, and partition function Q, giving the number of ways of writing the integer n as a sum of positive integers without regard to order and with the constraint that all integers in each sum are distinct. The partition function b, which gives the number of partitions of n in which no parts are multiples of k, is sometimes also used (Gordon and Ono 1997).

The Euler transform gives the number of partitions of n into integer parts of which there are different types of parts of size 1, of size 2, etc. For example, if for all n, then is the number of partitions of n into integer parts. Similarly, if for n prime and for n composite, then is the number of partitions of n into prime parts (Sloane and Plouffe 1995, p. 21).

A partition of a number n into a sum of elements of a list L can be determined using a greedy algorithm. The following table gives the number of partitions of n into a sum of positive powers p for multiples of n.

 n p = 1 p = 2 p = 3 p = 4 Sloane A000041 A001156 A003108 A046042 10 42 4 2 1 50 204226 104 10 4 100 190569292 1116 39 9 150 40853235313 6521 97 15 200 3972999029388 27482 208 24 250 230793554364681 388 34 300 9253082936723602 683 49 Andrews, G. E. The Theory of Partitions. Cambridge, England: Cambridge University Press, 1998.

Dickson, L. E. "Partitions." Ch. 3 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 101-164, 1952.

Gordon, B. and Ono, K. "Divisibility of Certain Partition Functions by Powers of Primes." Ramanujan J. 1, 25-34, 1997.

Hardy, G. H. and Wright, E. M. "Partitions." Ch. 19 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 273-296, 1979.

Savage, C. "Gray Code Sequences of Partitions." J. Algorithms 10, 577-595, 1989.

Skiena, S. "Partitions." §2.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 51-59, 1990.

Sloane, N. J. A. Sequences A000041/M0663, A001156/M0221, A003108/M0209, and A046042 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

Author: Eric W. Weisstein    