§27.14 Unrestricted Partitions

A fundamental problem studies the number of ways $n$ can be written as a sum of positive integers $\le n$, that is, the number of solutions of

The number of summands is unrestricted, repetition is allowed, and the order of the summands is not taken into account. The corresponding unrestricted partition function is denoted by $p\left(n\right)$, and the summands are called parts; see §26.9(i). For example, $p\left(5\right)=7$ because there are exactly seven partitions of $5$: $5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1$.

The number of partitions of $n$ into at most $k$ parts is denoted by $p_k\left(n\right)$; again see §26.9(i).

Euler introduced the reciprocal of the infinite product

as a generating function for the function $p\left(n\right)$ defined in §27.14(i):

with $p\left(0\right)=1$. Euler’s pentagonal number theorem states that

where the exponents $1$, $2$, $5$, $7$, $12$, $15$, $\mathrm{\dots }$ are the pentagonal numbers, defined by

Multiplying the power series for $f\left(x\right)$ with that for $1/f\left(x\right)$ and equating coefficients, we obtain the recursion formula

where $p\left(k\right)$ is defined to be $0$ if . Logarithmic differentiation of the generating function $1/f\left(x\right)$ leads to another recursion:

where ${\sigma }_1\left(k\right)$ is defined by (27.2.10) with $\alpha =1$.

These recursions can be used to calculate $p\left(n\right)$, which grows very rapidly. For example, $p\left(10\right)=42,p\left(100\right)$ = $\mathrm{1905\hspace{0.33em}69292}$, and $p\left(200\right)=\mathrm{397\hspace{0.33em}29990\hspace{0.33em}29388}$. For large $n$

where $K=\pi \sqrt{2/3}$ (Hardy and Ramanujan (1918)). Rademacher (1938) derives a convergent series that also provides an asymptotic expansion for $p\left(n\right)$:

where

and $s(h,k)$ is a Dedekind sum given by

Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta \left(\tau \right)$, defined by

This is related to the function $f\left(x\right)$ in (27.14.2) by

$\eta \left(\tau \right)$ satisfies the following functional equation: if $a,b,c,d$ are integers with $ad-bc=1$ and $c>0$, then

where $\epsilon =\exp \left(\pi \mathrm{i}(((a+d)/(12c))-s(d,c))\right)$ and $s(d,c)$ is given by (27.14.11).

For further properties of the function $\eta \left(\tau \right)$ see §§23.15–23.19.

Ramanujan (1921) gives identities that imply divisibility properties of the partition function. For example, the Ramanujan identity

implies $p\left(5n+4\right)\equiv 0(mod5)$. Ramanujan also found that $p\left(7n+5\right)\equiv 0(mod7)$ and $p\left(11n+6\right)\equiv 0(mod11)$ for all $n$. After decades of nearly fruitless searching for further congruences of this type, it was believed that no others existed, until it was shown in Ono (2000) that there are infinitely many. Ono proved that for every prime $q>3$ there are integers $a$ and $b$ such that $p\left(an+b\right)\equiv 0(modq)$ for all $n$. For example, $p\left(\mathrm{1575\hspace{0.33em}25693}n+\mathrm{1\hspace{0.33em}11247}\right)\equiv 0(mod13)$.

The discriminant function $\mathrm{\Delta }\left(\tau \right)$ is defined by

and satisfies the functional equation

if $a,b,c,d$ are integers with $ad-bc=1$ and $c>0$.

The 24th power of $\eta \left(\tau \right)$ in (27.14.12) with ${\mathrm{e}}^{2\pi \mathrm{i}\tau }=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau \left(n\right)$:

The tau function is multiplicative and satisfies the more general relation:

Lehmer (1947) conjectures that $\tau \left(n\right)$ is never 0 and verifies this for all by studying various congruences satisfied by $\tau \left(n\right)$, for example:

For further information on partitions and generating functions see Andrews (1976); also §§17.2–17.14, and §§26.9–26.10.