partition function

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1: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function $p\left(n\right)$ for . …To compute a particular value $p\left(n\right)$ it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). …

2: 26.9 Integer Partitions: Restricted Number and Part Size

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26.9.1 $$p_k\left(n\right)=p\left(n\right),$$ $k\ge n$.

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Symbols:

$p\left(n\right)$: total number of partitions of $n$ , $p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(i), §26.9 and Ch.26

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Table 26.9.1: Partitions $p_k\left(n\right)$.

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$n$	$k$
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$p_1\left(n\right)=1,$

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26.9.8 $$p_k\left(n\right)=p_k\left(n-k\right)+p_{k-1}\left(n\right);$$

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Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§26.10(iii)

Permalink:

http://dlmf.nist.gov/26.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(iii), §26.9 and Ch.26

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3: 26.2 Basic Definitions

… ►The total number of partitions of $n$ is denoted by $p\left(n\right)$. … ►

Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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4: 27.14 Unrestricted Partitions

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§27.14(i) Partition Functions

… ►as a generating function for the function $p\left(n\right)$ defined in §27.14(i): …with $p\left(0\right)=1$. … ►where $p\left(k\right)$ is defined to be $0$ if . … ►For example, $p\left(10\right)=42,p\left(100\right)$ = $\mathrm{1905\hspace{0.33em}69292}$, and $p\left(200\right)=\mathrm{397\hspace{0.33em}29990\hspace{0.33em}29388}$. …

5: 26.17 The Twelvefold Way

… ►In this table ${\left(k\right)}_n$ is Pochhammer’s symbol, and $S(n,k)$ and $p_k\left(n\right)$ are defined in §§26.8(i) and 26.9(i). … ►

Table 26.17.1: The twelvefold way.

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elements of $N$	elements of $K$	$f$ unrestricted	$f$ one-to-one	$f$ onto
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unlabeled	unlabeled	$p_k\left(n\right)$		$p_k\left(n\right)-p_{k-1}\left(n\right)$

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6: 26.21 Tables

… ►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. …

7: 27.21 Tables

… ►The partition function $p\left(n\right)$ is tabulated in Gupta (1935, 1937), Watson (1937), and Gupta et al. (1958). …

8: 26.1 Special Notation

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$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
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$p\left(n\right)$	number of partitions of $n$.
$p_k\left(n\right)$	number of partitions of $n$ into at most $k$ parts.
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9: 5.20 Physical Applications

… ►Then the partition function (with $\beta =1/(kT)$) is given by … ►and the partition function is given by ►

5.20.5 $${\psi }_n(\beta )=\frac{1}{{(2\pi )}^n}{\int }_{{[-\pi ,\pi ]}^n}{\mathrm{e}}^{-\beta W}d{\theta }_1\mathrm{\cdots }d{\theta }_n=\mathrm{\Gamma }\left(1+\frac{1}{2}n\beta \right){(\mathrm{\Gamma }\left(1+\frac{1}{2}\beta \right))}^{-n}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $n$: nonnegative integer, $W$: potential energy, ${\psi }_n(\beta )$: partition function and $\beta =1/(kT)$

Permalink:

http://dlmf.nist.gov/5.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

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Elementary Particles

… ►Carlitz (1972) describes the partition function of dense hadronic matter in terms of a gamma function. …

10: 25.17 Physical Applications

… ►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …