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1: 26.19 Mathematical Applications

§26.19 Mathematical Applications

… ►Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). …

2: 26.21 Tables

§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. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. …

3: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

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§26.9(i) Definitions

… ►Unrestricted partitions are covered in §27.14. … ►

§26.9(ii) Generating Functions

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§26.9(iii) Recurrence Relations

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4: 26.2 Basic Definitions

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Partition

… ►As an example, $\{1,3,4\}$, $\{2,6\}$, $\{5\}$ is a partition of $\{1,2,3,4,5,6\}$. … ►The total number of partitions of $n$ is denoted by $p\left(n\right)$. …For the actual partitions ($\pi$) for $n=1(1)5$ see Table 26.4.1. ►The integers whose sum is $n$ are referred to as the parts in the partition. …

5: 26.10 Integer Partitions: Other Restrictions

§26.10 Integer Partitions: Other Restrictions

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§26.10(i) Definitions

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§26.10(ii) Generating Functions

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§26.10(iv) Identities

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6: 26.20 Physical Applications

… ►The latter reference also describes chemical applications of other combinatorial techniques. ►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). …

7: 26.12 Plane Partitions

§26.12 Plane Partitions

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§26.12(i) Definitions

… ►Different configurations are counted as different plane partitions. … ► … ►The plane partition in Figure 26.12.1 is an example of a cyclically symmetric plane partition. …

8: 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). …

9: 27.14 Unrestricted Partitions

§27.14 Unrestricted Partitions

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§27.14(i) Partition Functions

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§27.14(iii) Asymptotic Formulas

… ►For example, $p\left(10\right)=42,p\left(100\right)$ = $\mathrm{1905\hspace{0.33em}69292}$, and $p\left(200\right)=\mathrm{397\hspace{0.33em}29990\hspace{0.33em}29388}$. … ►

§27.14(v) Divisibility Properties

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10: 26.1 Special Notation

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$x$	real variable.
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$\lambda$	integer partition.
$\pi$	plane partition.
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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 $n$.
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$\mathrm{pp}\left(n\right)$	number of plane partitions of $n$.
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