26 Combinatorial AnalysisProperties26.6 Other Lattice Path Numbers26.8 Set Partitions: Stirling Numbers

§26.7 Set Partitions: Bell Numbers

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Permalink:

http://dlmf.nist.gov/26.7

See also:

Annotations for Ch.26

Contents

1. §26.7(i) Definitions
2. §26.7(ii) Generating Function
3. §26.7(iii) Recurrence Relation
4. §26.7(iv) Asymptotic Approximation

§26.7(i) Definitions

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Defines:

$B\left(n\right)$: Bell number

Keywords:

Bell numbers, definition

Notes:

See Comtet (1974, pp. 210–211). (26.7.3) is from Wilf (1994, p. 22). Table 26.7.1 was computed by the author.

Permalink:

http://dlmf.nist.gov/26.7.i

See also:

Annotations for §26.7 and Ch.26

$B\left(n\right)$ is the number of partitions of $\{1,2,\mathrm{\dots },n\}$. For $S(n,k)$ see §26.8(i).

26.7.1		$$B\left(0\right)=1,$$
ⓘ Symbols: $B\left(n\right)$: Bell number Permalink: http://dlmf.nist.gov/26.7.E1 Encodings: TeX, pMML, png See also: Annotations for §26.7(i), §26.7 and Ch.26

26.7.2		$$B\left(n\right)=\sum _{k=0}^{n}S(n,k),$$
ⓘ Symbols: $B\left(n\right)$: Bell number, $S(n,k)$: Stirling number of the second kind, $k$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/26.7.E2 Encodings: TeX, pMML, png See also: Annotations for §26.7(i), §26.7 and Ch.26

26.7.3		$$B\left(n\right)=\sum _{k=1}^m\frac{k^n}{k!}\sum _{j=0}^{m-k}\frac{{(-1)}^j}{j!},$$
$m\ge n$,
ⓘ Symbols: $B\left(n\right)$: Bell number, $!$: factorial (as in $n!$), $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §26.7(i) Permalink: http://dlmf.nist.gov/26.7.E3 Encodings: TeX, pMML, png See also: Annotations for §26.7(i), §26.7 and Ch.26

26.7.4		$$B\left(n\right)={\mathrm{e}}^{-1}\sum _{k=1}^{\mathrm{\infty }}\frac{k^n}{k!}=1+\lfloor {\mathrm{e}}^{-1}\sum _{k=1}^{2n}\frac{k^n}{k!}\rfloor .$$
ⓘ Symbols: $B\left(n\right)$: Bell number, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$, $k$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/26.7.E4 Encodings: TeX, pMML, png See also: Annotations for §26.7(i), §26.7 and Ch.26

See Table 26.7.1.

$n$	$B\left(n\right)$	$n$	$B\left(n\right)$
0	1	10	1 15975
1	1	11	6 78570
2	2	12	42 13597
3	5	13	276 44437
4	15	14	1908 99322
5	52	15	13829 58545
6	203	16	1 04801 42147
7	877	17	8 28648 69804
8	4140	18	68 20768 06159
9	21147	19	583 27422 05057

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Symbols:

$B\left(n\right)$: Bell number and $n$: nonnegative integer

Keywords:

Bell numbers, table

Referenced by:

§26.7(i), §26.7(i)

Permalink:

http://dlmf.nist.gov/26.7.T1

See also:

Annotations for §26.7(i), §26.7 and Ch.26

§26.7(ii) Generating Function

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Keywords:

Bell numbers, generating function

Notes:

See Comtet (1974, p.210).

Permalink:

http://dlmf.nist.gov/26.7.ii

See also:

Annotations for §26.7 and Ch.26

26.7.5		$$\sum _{n=0}^{\mathrm{\infty }}B\left(n\right)\frac{x^n}{n!}=\exp ({\mathrm{e}}^x-1).$$
ⓘ Symbols: $B\left(n\right)$: Bell number, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $x$: real variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/26.7.E5 Encodings: TeX, pMML, png See also: Annotations for §26.7(ii), §26.7 and Ch.26

§26.7(iii) Recurrence Relation

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Keywords:

Bell numbers, recurrence relation

Notes:

See Comtet (1974, p.210).

Permalink:

http://dlmf.nist.gov/26.7.iii

See also:

Annotations for §26.7 and Ch.26

26.7.6		$$B\left(n+1\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B\left(k\right).$$
ⓘ Symbols: $B\left(n\right)$: Bell number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer and $n$: nonnegative integer Referenced by: Erratum (V1.0.1) for Equation (26.7.6) Permalink: http://dlmf.nist.gov/26.7.E6 Encodings: TeX, pMML, png Errata (effective with 1.0.1): Originally appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$: $$B\left(n+1\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B\left(n\right)$$ Reported 2010-11-07 by Layne Watson See also: Annotations for §26.7(iii), §26.7 and Ch.26

§26.7(iv) Asymptotic Approximation

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Keywords:

Bell numbers, asymptotic approximations

Notes:

See de Bruijn (1961, pp. 104–108) and Olver (1997b, pp. 329–331).

Referenced by:

Erratum (V1.0.21) for Subsection 26.7(iv), Erratum (V1.2.2) for Subsection 26.7(iv)

Permalink:

http://dlmf.nist.gov/26.7.iv

Modification (effective with 1.2.2):

Immediately below (26.7.8), $\mathrm{Wp}\left(n\right)$ was replaced with $W_0\left(n\right)$ twice and the text “Lambert function” was replaced with the text “Lambert $W$-function”.

Correction (effective with 1.0.21):

In the final line of this subsection, $\mathrm{Wm}\left(n\right)$ was replaced by $\mathrm{Wp}\left(n\right)$ twice. Also, the wording “or, equivalently, $N={\mathrm{e}}^{\mathrm{Wm}\left(n\right)}$” was changed to “or, specifically, $N={\mathrm{e}}^{\mathrm{Wp}\left(n\right)}$”.

Suggested 2018-04-09 by Gergő Nemes

See also:

Annotations for §26.7 and Ch.26

26.7.7		$$B\left(n\right)=\frac{N^n{\mathrm{e}}^{N-n-1}}{{(1+\ln N)}^{1/2}}\left(1+O\left(\frac{{(\ln n)}^{1/2}}{n^{1/2}}\right)\right),$$
$n\to \mathrm{\infty }$,
ⓘ Symbols: $B\left(n\right)$: Bell number, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $N$ Permalink: http://dlmf.nist.gov/26.7.E7 Encodings: TeX, pMML, png See also: Annotations for §26.7(iv), §26.7 and Ch.26

where

26.7.8		$$N\ln N=n,$$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $N$ Referenced by: §26.7(iv), Erratum (V1.2.2) for Subsection 26.7(iv) Permalink: http://dlmf.nist.gov/26.7.E8 Encodings: TeX, pMML, png See also: Annotations for §26.7(iv), §26.7 and Ch.26

or, specifically, $N={\mathrm{e}}^{W_0\left(n\right)}$, with properties of the Lambert $W$-function $W_0\left(n\right)$ given in §4.13. For higher approximations to $B\left(n\right)$ as $n\to \mathrm{\infty }$ see de Bruijn (1961, pp. 104–108).

26.6 Other Lattice Path Numbers26.8 Set Partitions: Stirling Numbers

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