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partition function∗
silverfish†
2013-03-21 20:38:16
The partition function p(n) is defined to be the number of partitions of the
integer n. The sequence of values p(0), p(1), p(2), . . . is Sloane’s A000041 and
begins 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, . . .. This function grows very quickly, as we
see in the following theorem due to Hardy and Ramanujan.
Theorem 1 As n → ∞, the ratio of p(n) and
√
eπ 2n/3
√
4n 3
approaches 1.
The generating function of p(n) is called F : by definition
F (x) =
∞
X
p(n)xn .
n=0
F can be written as an infinite product:
F (x) =
∞
Y
(1 − xi )−1 .
i=1
To see this, expand each term in the product as a power series:
∞
Y
(1 + xi + x2i + x3i + · · · ).
i=1
Now expand this as a power series. Given a partition of n with ai parts of size
i ≥ 1, we get a term xn in this expansion by choosing xa1 from the first term in
the product, x2a2 from the second, x3a3 from the third and so on. Clearly any
term xn in the expansion arises in this way from a partition of n.
∗ hPartitionFunctioni created: h2013-03-21i by: hsilverfishi version: h37980i Privacy
setting: h1i hDefinitioni h05A17i
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One can prove in the same way that the generating function Fm for the
number pm (n) of partitions of n into at most m parts (or equivalently into
parts of size at most m) is
Fm (x) =
m
Y
(1 − xi )−1 .
i=1
References
[1] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers,
Oxford University Press, 2003.
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