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1: Possible Errors in DLMF

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. There are also cases where browser bugs or poor fonts can be misleading; you can verify MathML display by comparing the to the images or TeXfound under Encodings in the Info boxes (see About MathML). …

2: Need Help?

… ►

• How do I search within DLMF? See Guide to Searching the DLMF.

• See also the Index or Notations sections.

• Links to definitions, keywords, annotations and other interesting information can be found in the Info boxes by clicking or hovering the mouse over the icon next to each formula, table, figure, and section heading.

3: 1 Algebraic and Analytic Methods

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4: About MathML

… ►For other browsers, you may see a ? or a box like indicating missing symbols, and thus insufficient fonts. …

5: How to Cite

… ►For convenience, the permalink can be found in the pop-up ‘Info box’ associated with each item in the site.

6: 26.18 Counting Techniques

… ►The number of ways of placing $n$ labeled objects into $k$ labeled boxes so that at least one object is in each box is …

7: 26.12 Plane Partitions

… ►We define the $r\times s\times t$ box $B(r,s,t)$ as ►

26.12.3 $$B(r,s,t)=\{(h,j,k)|\mathrm{\hspace{0.25em}1}\le h\le r,1\le j\le s,1\le k\le t\}.$$

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Symbols:

$r$: positive integer, $s$: positive integer, $t$: positive integer, $h$: positive integer, $j$: positive integer, $k$: positive integer and $B(r,s,t)$: Box

Permalink:

http://dlmf.nist.gov/26.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(i), §26.12 and Ch.26

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26.12.4 $$\prod _{(h,j,k)\in B(r,s,t)}\frac{h+j+k-1}{h+j+k-2}=\prod _{h=1}^r\prod _{j=1}^s\frac{h+j+t-1}{h+j-1}.$$

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Symbols:

$\in$: element of, $r$: positive integer, $s$: positive integer, $t$: positive integer, $h$: positive integer, $j$: positive integer, $k$: positive integer and $B(r,s,t)$: Box

Permalink:

http://dlmf.nist.gov/26.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(i), §26.12 and Ch.26

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26.12.21 $$\sum _{\pi \subseteq B(r,s,t)}{q}^{|\pi |}=\prod _{(h,j,k)\in B(r,s,t)}\frac{1-q^{h+j+k-1}}{1-q^{h+j+k-2}}=\prod _{h=1}^r\prod _{j=1}^s\frac{1-q^{h+j+t-1}}{1-q^{h+j-1}},$$

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Symbols:

$\in$: element of, $\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $k$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $s$: positive integer, $t$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.12.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

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26.12.22

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Symbols:

$\subseteq$: is, or is contained in, $h$: nonnegative integer, $j$: nonnegative integer, $\pi$: plane partition, $r$: positive integer, $t$: positive integer, $B(r,s,t)$: Box and $q$: parameter

Referenced by:

§26.12(ii)

Permalink:

http://dlmf.nist.gov/26.12.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(ii), §26.12 and Ch.26

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8: 26.17 The Twelvefold Way

… ►The twelvefold way gives the number of mappings $f$ from set $N$ of $n$ objects to set $K$ of $k$ objects (putting balls from set $N$ into boxes in set $K$). …

9: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ► $\left(\genfrac{}{}{0.0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_k}\right)$ is the number of ways of placing $n=n_1+n_2+\mathrm{\cdots }+n_k$ distinct objects into $k$ labeled boxes so that there are $n_j$ objects in the $j$th box. …

10: Guide to Searching the DLMF

… ►Every page contains a search box in the navigation bar. …