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11: 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
…
unlabeled	unlabeled	$p_k\left(n\right)$		$p_k\left(n\right)-p_{k-1}\left(n\right)$

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12: 17.16 Mathematical Applications

… ►Many special cases of $q$-series arise in the theory of partitions, a topic treated in §§27.14(i) and 26.9. …

13: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

§26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►Table 26.4.1 gives numerical values of multinomials and partitions $\lambda ,M_1,M_2,M_3$ for $1\le m\le n\le 5$. …$\lambda$ is a partition of $n$: …$M_3$ is the number of set partitions of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ subsets of size 1, $a_2$ subsets of size 2, $\mathrm{\dots }$, and $a_n$ subsets of size $n$: … ►

Table 26.4.1: Multinomials and partitions.

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$n$	$m$	$\lambda$	$M_1$	$M_2$	$M_3$
…

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…

14: 26.11 Integer Partitions: Compositions

§26.11 Integer Partitions: Compositions

►A composition is an integer partition in which order is taken into account. …

15: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
…

16: 5.20 Physical Applications

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

5.20.3 $${\psi }_n(\beta )={\int }_{{ℝ}^n}{\mathrm{e}}^{-\beta W}dx={(2\pi )}^{n/2}{\beta }^{-(n/2)-(\beta n(n-1)/4)}{\left(\mathrm{\Gamma }\left(1+\frac{1}{2}\beta \right)\right)}^{-n}\prod _{j=1}^n\mathrm{\Gamma }\left(1+\frac{1}{2}j\beta \right).$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/5.20.E3

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See also:

Annotations for §5.20, §5.20 and Ch.5

… ►and the partition function is given by … ►

Elementary Particles

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

17: 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)). …

18: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

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§35.4(i) Definitions

►A partition $\kappa =(k_1,\mathrm{\dots },k_m)$ is a vector of nonnegative integers, listed in nonincreasing order. … ►The partitional shifted factorial is given by … ►For any partition $\kappa$, the zonal polynomial $Z_{\kappa }:𝓢\to ℝ$ is defined by the properties …

19: 27.21 Tables

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

20: Bibliography Y

… ►

Partitions with difference conditions and Alder’s conjecture. Proc. Natl. Acad. Sci. USA 101 (47), pp. 16417–16418.

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External Links:

ISSN 1091-6490, FullText, Document, MathReview (Scott D. Ahlgren), ZentralBlatt

Referenced by:

§26.10(iv).

Encodings:

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