Stirling numbers (first and second kinds)

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1: 26.8 Set Partitions: Stirling Numbers

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Table 26.8.1: Stirling numbers of the first kind $s(n,k)$.

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$n$	$k$
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$s(n,0)=0,$

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$S(n,0)=0,$

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$S(n,1)=1,$

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2: 26.1 Special Notation

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$x$	real variable.
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$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
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$s(n,k)$	Stirling numbers of the first kind.
$S(n,k)$	Stirling numbers of the second kind.

… ►Other notations for $s(n,k)$, the Stirling numbers of the first kind, include $S_n^{(k)}$ (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), $S_n^k$ (Jordan (1939), Moser and Wyman (1958a)), $\left(\genfrac{}{}{0.0pt}{}{n-1}{k-1}\right)B_{n-k}^{(n)}$ (Milne-Thomson (1933)), ${(-1)}^{n-k}{S}_1(n-1,n-k)$ (Carlitz (1960), Gould (1960)), ${(-1)}^{n-k}\left[\genfrac{}{}{0pt}{}{n}{k}\right]$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for $S(n,k)$, the Stirling numbers of the second kind, include ${𝒮}_n^{(k)}$ (Fort (1948)), ${𝔖}_n^k$ (Jordan (1939)), ${\sigma }_n^k$ (Moser and Wyman (1958b)), $\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B_{n-k}^{(-k)}$ (Milne-Thomson (1933)), $S_2(k,n-k)$ (Carlitz (1960), Gould (1960)), $\left\{\genfrac{}{}{0pt}{}{n}{k}\right\}$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

3: 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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labeled	unlabeled	$\begin{array}{c}S(n,1)+S(n,2)\\ +\mathrm{\cdots }+S(n,k)\end{array}$		$S(n,k)$
…

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4: 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. … ►It also contains a table of Gaussian polynomials up to ${\left[\genfrac{}{}{0.0pt}{}{12}{6}\right]}_q$. …

5: 26.13 Permutations: Cycle Notation

… ►The Stirling cycle numbers of the first kind, denoted by $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$, count the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. They are related to Stirling numbers of the first kind by ►

26.13.3 $$\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]=\left|s(n,k)\right|.$$

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Defines:

$\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind

Symbols:

$s(n,k)$: Stirling number of the first kind, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§4.13

Permalink:

http://dlmf.nist.gov/26.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.13 and Ch.26

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6: 24.15 Related Sequences of Numbers

… ►The Stirling numbers of the first kind $s(n,m)$, and the second kind $S(n,m)$, are as defined in §26.8(i). … ►

24.15.7 $$B_n=\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n+1}{k+1}\right)S(n+k,k)/\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right),$$

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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24.15.8 $$\sum _{k=0}^n{(-1)}^{n+k}s(n+1,k+1)B_k=\frac{n!}{n+1}.$$

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Symbols:

$B_n$: Bernoulli numbers, $s(n,k)$: Stirling number of the first kind, $!$: factorial (as in $n!$), $k$: integer and $n$: integer

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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24.15.9 $$p\frac{B_n}{n}\equiv S(p-1+n,p-1)(mod{p}^2),$$ $1\le n\le p-2$,

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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24.15.10 $$\frac{2n-1}{4n}{p}^2{B}_{2n}\equiv S(p+2n,p-1)(mod{p}^3),$$ $2\le 2n\le p-3$.

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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7: 26.7 Set Partitions: Bell Numbers

… ►For $S(n,k)$ see §26.8(i). … ►

26.7.2 $$B\left(n\right)=\sum _{k=0}^{n}S(n,k),$$

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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:

pMML, png, TeX

See also:

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

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8: 4.13 Lambert $W$-Function

… ►For the definition of Stirling cycle numbers of the first kind $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$ see (26.13.3). … ►

4.13.10 $$W_k\left(z\right)\sim {\xi }_k-\ln {\xi }_k+\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{{\xi }_k^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln {\xi }_k\right)}^m}{m!},$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $k$: integer, $m$: integer, $n$: integer and $z$: complex variable

Notes:

See Corless et al. (1996, p. 350) and Jeffrey et al. (1995, Theorem 2).

Referenced by:

§4.13, §4.13, §4.45(iii), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

Originally we gave only the first three terms of the infinite series.

See also:

Annotations for §4.13 and Ch.4

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4.13.11 $$W_{\pm 1}\left(x\mp 0\mathrm{i}\right)\sim -\eta -\ln \eta +\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\eta }^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln \eta \right)}^m}{m!},$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $m$: integer, $n$: integer, $x$: real variable and $\eta$

Notes:

See Corless et al. (1996, p. 350).

Referenced by:

§4.13, Erratum (V1.1.2) for Equation (4.13.11), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E11

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

Originally we gave only the first three terms of the infinite series.

See also:

Annotations for §4.13 and Ch.4

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9: 26.14 Permutations: Order Notation

… ►In this subsection $S(n,k)$ is again the Stirling number of the second kind (§26.8), and $B_m$ is the $m$th Bernoulli number (§24.2(i)). … ►

26.14.7

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Symbols:

: Eulerian number, $S(n,k)$: Stirling number of the second kind, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $j$: nonnegative integer, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(iii), §26.14 and Ch.26

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26.14.12 $n\ge m$, $n\ge 1$.

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Symbols:

: Eulerian number, $S(n,k)$: Stirling number of the second kind, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.14.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(iii), §26.14 and Ch.26

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10: 24.16 Generalizations

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24.16.8 $${\beta }_n(\lambda )=n!b_n{\lambda }^n+\sum _{k=1}^{\lfloor n/2\rfloor }\frac{n}{2k}{B}_{2k}s(n-1,2k-1){\lambda }^{n-2k},$$ $n=2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $s(n,k)$: Stirling number of the first kind, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $k$: integer, $n$: integer and ${\beta }_n(\lambda )$

Referenced by:

§24.16(i)

Permalink:

http://dlmf.nist.gov/24.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

►Here $s(n,m)$ again denotes the Stirling number of the first kind. …