Partition function (number theory)

In number theory the partition function p(n) counts the number of partitions of a positive integer n, that is, the number of ways of expressing n as a sum of positive integers (where order is not significant).

Thus p(3) = 3, since the number 3 has 3 partitions:

• 3
• 2+1
• 1+1+1

Properties

The partition function satisfies an asymptotic relation

$p(n)\sim {\frac {\exp \left(\pi {\sqrt {2/3}}{\sqrt {n}}\right)}{4n{\sqrt {3}}}}.$

References

• Tom M. Apostol (1990). Modular functions and Dirichlet Series in Number Theory, 2nd ed. Springer-Verlag, 94-112. ISBN 0-387-97127-0. 
• G.H. Hardy; E. M. Wright (2008). An Introduction to the Theory of Numbers, 6th ed.. Oxford University Press, 361-392. ISBN 0-19-921986-5.