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Partition (number theory)

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In number theory, a partition of a positive integer n is a way of writing n as a sum of positive integers. Two sums which only differ in the order of their summands are considered to be the same partition; if order matters then the sum becomes a composition. A summand in a partition is also called a part. The number of partitions of n is given by the partition function p(n).

Contents

Examples

The partitions of 4 are listed below:

The partitions of 8 are listed below:

Partition function

In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers. For example, 4 can be partitioned in 5 distinct ways:

4,     3 + 1,     2 + 2,     2 + 1 + 1,     1 + 1 + 1 + 1.
So p(4) = 5. By convention p(0) = 1, p(n) = 0 for n negative. Partitions can be graphically visualized with Young diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and in group representation theory in general.

Intermediate function

One way of getting a handle on the partition function involves an intermediate function p(k, n) which represents the number of partitions of n using only natural numbers at least as large as k. For any given value of k, partitions counted by p(k, n) fit into exactly one of the following categories:

  1. smallest addend is k
  2. smallest addend is strictly greater than k
The number of partitions meeting the first condition is p(k, n − k). To see this, imagine a list of all the partitions of the number n − k into numbers of size at least k, then imagine appending "+ k" to each partition in the list. Now what is it a list of?

The number of partitions meeting the second condition is p(k + 1, n) since a partition into parts of at least k which contains no parts of exactly k must have all parts at least k + 1.

Since the two conditions are mutually exclusive, the number of partitions meeting either condition is p(k + 1, n) + p(k, n − k). The base cases of this recursively defined function are as follows:

This function tends to exhibit deceptive behavior.

p(1, 4) = 5
p(2, 8) = 7
p(3, 12) = 9
p(4, 16) = 11
p(5, 20) = 13
p(6, 24) = 16
Our original function p(n) is just p(1, n).

The values of this function:

k
1 2 3 4 5 6 7 8 9 10
n 1 1
2 2 1
3 3 1 1
4 5 2 1 1
5 7 2 1 1 1
6 11 4 2 1 1 1
7 15 4 2 1 1 1 1
8 22 7 3 2 1 1 1 1
9 30 8 4 2 1 1 1 1 1
10 42 12 5 3 2 1 1 1 1 1

Generating function

A generating function for p(n) is given by the reciprocal of Euler's function:

[\sum_^\infty p(n)x^n = \prod_^\infty \left(\frac \right).]
Expanding each term on the right-hand side as a geometric series, we can rewrite it as

(1 + x + x2 + x3 + ...)(1 + x2 + x4 + x6 + ...)(1 + x3 + x6 + x9 + ...) ...
The xn term in this product counts the number of ways to write

n = a1 + 2a2 + 3a3 + ... = (1 + 1 + ... + 1) + (2 + 2 + ... + 2) + (3 + 3 + ... + 3) + ...,
where each number i appears ai times. This is precisely the definition of a partition of n, so our product is the desired generating function. More generally, the generating function for the partitions of n into numbers from a set A can be found by taking only those terms in the product where k is an element of A. This result is due to Euler.

The formulation of Euler's generating function is a special case of a q-series and is similar to the product formulation of many modular forms, and specifically the Dedekind eta function. It can also be used in conjunction with the pentagonal number theorem to derive a recurrence for the partition function stating that:

p(k) − p(k − 1) − p(k − 2) + p(k − 5) + p(k − 7) − p(k − 12) − ... = 0,
where the sum is taken over all pentagonal numbers of the form ½n(3n − 1), including those where n < 0, and the terms continue to alternate +, +, −, −, +, +, ...

Table of values

Some values of the partition function are as follows:

  • p(1) = 1
  • p(2) = 2
  • p(3) = 3
  • p(4) = 5
  • p(5) = 7
  • p(6) = 11
  • p(7) = 15
  • p(8) = 22
  • p(9) = 30
  • p(10) = 42
  • p(100) = 190,569,292
  • p(1000) = 24,061,467,864,032,622,473,692,149,727,991 [2.4 × 1031]

Rademacher's series

An asymptotic expression for p(n) is given by

[p(n) \sim \frac \right) } } \mbox n\rightarrow \infty.]
This expression was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920. Considering [p(1000)], the asymptotic formula gives about [2.4402\times 10^], reasonably close to the correct answer given above.

In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan's results by providing a convergent series expression for p(n). It is

[p(n)=\frac} \sum_^\infty A_k(n)\;\sqrt \; \frac\left( \frac \sqrt\left(n-\frac\right)}\right) }}}\right)]
where
[A_k(n) = \sum_\exp \left( \pi i s(m,k) - 2\pi inm/k \right).]
Here, the notation [(m,n)=1] implies that the sum should occur only over the values of m that are relatively prime to n. The function [s(m,k)] is a Dedekind sum. The proof of Rademacher's formula is interesting in that it involves Ford circles, Farey sequences, modular symmetry and the Dedekind eta function in a central way.

Congruences

Mathematician Srinivasa Ramanujan is credited with discovering that "congruences" in the number of partitions exist for integers ending in 4 and 9. Starting at the number 4, the partition number for every 5th integer is divisible by 5. For instance, the number of partitions for the integer 4 is 5. For the integer 9, the number of partitions is 30; for 14 there are 135 partitions. The integers 5, 30, and 135 are all evenly divisible by 5. Ramanujan demonstrated that this congruence goes indefinitely. He also discovered congruences related to 7 and 11. Starting with 5, the partition number for every 7th integer (5, 12, 19, ... ) is itself divisible by 7. Starting with 6, the partition number for every 11th integer is divisible by 11. Since 5, 7, and 11 are consecutive primes, one might think that starting with 7, the partition number of every 13th integer is divisible by 13. This is, however, false.

In more recent times, A. O. L. Atkin of the University of Illinois at Chicago discovered additional congruences. For example, the partition number for every 17,303rd integer starting with 237 is divisible by 13.

Restricted partitions

Among the 22 partitions for the number 8, 6 contain only odd parts:
  • 7 + 1
  • 5 + 3
  • 5 + 1 + 1 + 1
  • 3 + 3 + 1 + 1
  • 3 + 1 + 1 + 1 + 1 + 1
  • 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
Curiously, if we count the partitions of 8 with distinct parts, we also obtain the number 6:
  • 8
  • 7 + 1
  • 6 + 2
  • 5 + 3
  • 5 + 2 + 1
  • 4 + 3 + 1
Is this only coincidence, or is it true that, for all positive numbers, the number of partitions with odd parts always equals the number of partitions with distinct parts? This and other results can be obtained by the aid of a visual tool, a Ferrers graph (also called Ferrers diagram, since it is not a graph in the graph-theoretical sense, or sometimes Young diagram, alluding to the Young tableau).

Ferrers graph

The partition 6 + 4 + 3 + 1 of the positive number 14 can be represented by the following graph, which is named in honor of Norman Macleod Ferrers:






6+4+3+1

The 14 circles are lined up in 4 columns, each having the size of a part of the partition. The graphs for the 5 partitions of the number 4 are listed below:





 







 






 






 







 4 
 =
 3+1 
 =
 2+2 
 =
 2+1+1 
 =
 1+1+1+1



If we now flip the graph of the partition 6 + 4 + 3 + 1 along the 
NW-SE axis, we obtain another partition of 14:




 




















 ↔
 



















 6+4+3+1
 =
 4+3+3+2+1+1



By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 +
1 + 1 of the number 14.  Such partitions are said to be 
conjugate of one another.   In the case of the number 4, partitions
4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1
are conjugate of each other.  Of particular interest is the partition
2 + 2, which has itself as conjugate.  Such a partition is said to be
self-conjugate.


Claim: The number of self-conjugate partitions is the same as the
number of partitions with distinct odd parts.


Proof (sketch): The crucial observation is that every odd part can
be "folded" in the middle to form a self conjugate graph:




 










 ↔
 










One can then obtain a bijection between the set of partitions with distinct
odd parts and the set of self-conjugate partitions, as illustrated by
the following example:




 




























 ↔
 


























 9+7+3
 =
 5+5+4+3+2


 dist. odd

 self-conjugate



Similar techniques can be employed to establish, for example, the following
equalities:

  • The number of partitions of n into no more than k parts is the same as the number of partitions of n into parts no larger than k.
  • The number of partitions of n into no more than k parts is the same as the number of partitions of n+k into exactly k parts.


Number of partitions


The number of partitions of a positive integer n is given by the partition function p(n).  The number of partitions of n into
exactly k parts is denoted by pk(n).


Ferrers graph techniques also allow us to prove results like the following:

  • There are p(n) − p(n − 1) partitions of n in which each part is at least 2.
  • p(1) + p(2) + ... + p(n) < p(2n)


See also



References


  • Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 (See chapter 5 for a modern pedagogical intro to Rademacher's formula).


  • D. H. Lehmer, On the remainder and convergence of the series for the partition function Trans. Amer. Math. Soc. 46(1939) pp 362-373. (Provides the main formula (no derivatives), remainder, and older form for Ak(n).)


  • Gupta, Gwyther, Miller, Roy. Soc. Math. Tables, vol 4, Tables of partitions, (1962) (Has text, nearly complete bibliography, but they (and Abramowitz) missed the Selberg formula for Ak(n) which is in Whiteman.)



  • Hans Rademacher, Collected Papers of Hans Rademacher, (1974) MIT Press; v II, p 100-107, 108-122, 460-475.

An elementary introduction to the topic of integer partition, including
a discussion of Ferrers graphs, can be found in the following reference:

Miklós Bóna, A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, World Scientific Publishing, 2002.  ISBN 9810249004.



Another highly accessible introduction is:

George Andrews and Kimmo Eriksson, Integer Partitions, Cambridge University Press, 2004. ISBN 0521600901.




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