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1: 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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2: 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. …

3: 26.12 Plane Partitions

§26.12 Plane Partitions

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§26.12(i) Definitions

… ►A strict shifted plane partition is an arrangement of the parts in a partition so that each row is indented one space from the previous row and there is weak decrease across rows and strict decrease down columns. … ►A descending plane partition is a strict shifted plane partition in which the number of parts in each row is strictly less than the largest part in that row and is greater than or equal to the largest part in the next row. The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition. …

4: 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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5: 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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6: 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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7: 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
labeled	labeled	$k^n$	${\left(k-n+1\right)}_n$	$k!S(n,k)$
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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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8: 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)). …

9: 26.1 Special Notation

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$x$	real variable.
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$\lambda$	integer 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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… ►Many combinatorics references use the rising and falling factorials: …

10: 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. …