gives the number of ways of writing the Integer as a sum of Positive Integers without regard to order. For example, since 4 can be written

(1) |

so . satisfies

(2) |

50 | 204226 |

100 | 190569292 |

200 | 3972999029388 |

300 | 9253082936723602 |

400 | 6727090051741041926 |

500 | 2300165032574323995027 |

600 | 458004788008144308553622 |

700 | 60378285202834474611028659 |

800 | 5733052172321422504456911979 |

900 | 415873681190459054784114365430 |

1000 | 24061467864032622473692149727991 |

for which is Prime are 2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, ... (Sloane's A046063). Numbers which cannot be written as a Product of are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (Sloane's A046064), corresponding to numbers of nonisomorphic Abelian Groups which are not possible for any group order.

When explicitly listing the partitions of a number , the simplest form is the so-called *natural representation*
which simply gives the sequence of numbers in the representation (e.g., (2, 1, 1) for the number ). The *multiplicity representation* instead gives the number of times each number occurs together with that number (e.g., (2, 1),
(1, 2) for
). The Ferrers Diagram is a pictorial representation of a partition.

Euler invented a Generating Function which gives rise to a Power Series in ,

(3) |

(4) |

(5) | |||

(6) |

where the exponents are generalized Pentagonal Numbers 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, ... (Sloane's A001318) and the sign of the th term (counting 0 as the 0th term) is (with the Floor Function), the partition numbers are given by the Generating Function

(7) |

(8) |

In 1916-1917, Hardy and Ramanujan used the Circle Method and elliptic Modular Functions to obtain the approximate solution

(9) |

(10) |

(11) | |||

(12) | |||

(13) | |||

(14) | |||

(15) | |||

(16) |

is the Floor Function, and runs through the Integers less than and Relatively Prime to (when , ). The remainder after terms is

(17) |

With as defined above, Ramanujan also showed that

(18) |

(19) |

(20) |

(21) |

Let be the number of partitions of containing Odd numbers only and be the number of partitions of without duplication, then

(22) |

Let be the number of partitions of Even numbers only, and let () be the number of partitions in which the parts are all Even (Odd) and all different. The first few values of are 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, ... (Sloane's A000700). Some additional Generating Functions are given by Honsberger (1985, pp. 241-242)

(23) | |

(24) | |

(25) | |

(26) | |

(27) | |

(28) |

- 1. The number of partitions of in which no Even part is repeated is the same as the number of partitions of in which no part occurs more than three times and also the same as the number of partitions in which no part is divisible by four.
- 2. The number of partitions of in which no part occurs more often than times is the same as the number of partitions in which no term is a multiple of .
- 3. The number of partitions of in which each part appears either 2, 3, or 5 times is the same as the number of partitions in which each part is Congruent mod 12 to either 2, 3, 6, 9, or 10.
- 4. The number of partitions of in which no part appears exactly once is the same as the number of partitions of in which no part is Congruent to 1 or 5 mod 6.
- 5. The number of partitions in which the parts are all Even and different is equal to the absolute difference of the number of partitions with Odd and Even parts.

, also written , is the number of ways of writing as a sum of terms, and can be computed from the
Recurrence Relation

(29) |

The function can be given explicitly for the first few values of ,

(30) | |||

(31) |

where is the Floor Function and is the Nint function (Honsberger 1985, pp. 40-45).

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Unrestricted Partitions.'' §24.2.1 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 825, 1972.

Adler, H. ``Partition Identities--From Euler to the Present.'' *Amer. Math. Monthly* **76**, 733-746, 1969.

Adler, H. ``The Use of Generating Functions to Discover and Prove Partition Identities.'' *Two-Year College Math. J.* **10**,
318-329, 1979.

Andrews, G. *Encyclopedia of Mathematics and Its Applications, Vol. 2: The Theory of Partitions.*
Cambridge, England: Cambridge University Press, 1984.

Berndt, B. C. *Ramanujan's Notebooks, Part IV.* New York: Springer-Verlag, 1994.

Conway, J. H. and Guy, R. K. *The Book of Numbers.* New York: Springer-Verlag, pp. 94-96, 1996.

Honsberger, R. *Mathematical Gems III.* Washington, DC: Math. Assoc. Amer., pp. 40-45 and 64-68, 1985.

Honsberger, R. *More Mathematical Morsels.* Washington, DC: Math. Assoc. Amer., pp. 237-239, 1991.

Jackson, D. and Goulden, I. *Combinatorial Enumeration.* New York: Academic Press, 1983.

MacMahon, P. A. *Combinatory Analysis.* New York: Chelsea, 1960.

Rademacher, H. ``On the Partition Function .'' *Proc. London Math. Soc.* **43**, 241-254, 1937.

Ruskey, F. ``Information of Numerical Partitions.'' http://sue.csc.uvic.ca/~cos/inf/nump/NumPartition.html.

Sloane, N. J. A. Sequences A000009/M0281, A000041/M0663, A000700/M0217, A001318/M1336, A046063, and A046064 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9

1999-05-26