Partition function (number theory)

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 recursive function are as follows:

• p(k, n) = 0 if k > n

• p(k, n) = 1 if k = n

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

${\displaystyle \sum _{n=0}^{\infty }p(n)x^{n}=\prod _{k=1}^{\infty }\left({\frac {1}{1-x^{k}}}\right)}$ 

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 x^m term in this product counts the number of ways to write

n = a₁ + 2a₂ + 3a₃ + ... = (1 + 1 + 1 + 1 + ...) + (2 + 2 + 2 + 2 + ...) + (3 + 3 + 3 ...) + ...,

where each number i appears a_i 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) = 190569292
• p(1000) = 24061467864032622473692149727991

Rademacher's series

An asymptotic expression for p(n) is given by

${\displaystyle p(n)\sim {\frac {\exp \pi {\sqrt {2n/3}}}{4n{\sqrt {3}}}}{\mbox{ as }}n\rightarrow \infty }$ 

This expression was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920.

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

${\displaystyle p(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }A_{k}(n)\;{\sqrt {k}}\;{\frac {d}{dn}}\left({\frac {\sinh \left({\frac {\pi }{k}}{\sqrt {{\frac {2}{3}}\left(n-{\frac {1}{24}}\right)}}\right)}{\sqrt {n-{\frac {1}{24}}}}}\right)}$ 

where

${\displaystyle A_{k}(n)=\sum _{0\leq m<k\;;\;(m,k)=1}\exp \left(\pi is(m,k)-2\pi inm/k\right)}$ .

Here, the notation ${\displaystyle (m,n)=1}$  implies that the sum should occur only over the values of m that are relatively prime to n. The function ${\displaystyle s(m,k)}$  is a Dedekind sum. The proof of Rademachers formula is interesting in that it involves Ford circles, Farey sequences, modular symmetry and the Dedekind eta function in a central way.

Congruences

For any number that ends in 4 or 9, the number of partitions is always divisible by 5.

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).

• Lehmer, D. H.; On the remainder and convergence of the series for the partition function;

Trans. Amer. Math. Soc. 46(1939) 362-373.

Main formula (no derivatives), remainder, but older form for A_k(n).

• Roy. Soc. Math. Tables, vol 4, Tables of partitions, 1962;

Gupta, Gwyther, Miller.

Has text, nearly complete bibliography,

but they (and Abramowitz) missed the Selberg formula for A_k(n) which is in:

• Whiteman, A. L.; A sum connected with the series for the partition function;

Pacific Journal of Math. 6(1956) 159-176.

Selberg formula. The older form is the finite Fourier expansion of Selberg.

• Rademacher's contributions are in: Collected Papers of Hans Rademacher;

MIT Press, 1974; v II, p 100-107, 108-122, 460-475.

External links:

• Partition and composition calculator
• First 4096 values of the partition function
• An algorithm to compute the partition function
• Details in Math World