# Stirling numbers (first and second kinds)

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##### 1: 26.8 Set Partitions: Stirling Numbers
$s\left(n,0\right)=0,$
$S\left(n,0\right)=0,$
$S\left(n,1\right)=1,$
##### 2: 26.1 Special Notation
 $x$ real variable. …
 $\genfrac{(}{)}{0.0pt}{}{m}{n}$ binomial coefficient. … Stirling numbers of the first kind. Stirling numbers of the second kind.
Other notations for $s\left(n,k\right)$, 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)), $\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}$ (Milne-Thomson (1933)), $(-1)^{n-k}S_{1}(n-1,n-k)$ (Carlitz (1960), Gould (1960)), $(-1)^{n-k}\left[n\atop k\right]$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). Other notations for $S\left(n,k\right)$, the Stirling numbers of the second kind, include $\mathscr{S}^{(k)}_{n}$ (Fort (1948)), $\mathfrak{S}_{n}^{k}$ (Jordan (1939)), $\sigma_{n}^{k}$ (Moser and Wyman (1958b)), $\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}$ (Milne-Thomson (1933)), $S_{2}(k,n-k)$ (Carlitz (1960), Gould (1960)), $\left\{n\atop k\right\}$ (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 $S\left(n,k\right)$ and $p_{k}\left(n\right)$ are defined in §§26.8(i) and 26.9(i). …
##### 4: 26.21 Tables
###### §26.21 Tables
Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\genfrac{(}{)}{0.0pt}{}{m}{n}$ 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, $s\left(n,k\right)$ and $S\left(n,k\right)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p\left(\mathcal{D},n\right)$ for $n$ up to 500. … It also contains a table of Gaussian polynomials up to $\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}$. …
##### 5: 24.15 Related Sequences of Numbers
The Stirling numbers of the first kind $s\left(n,m\right)$, and the second kind $S\left(n,m\right)$, are as defined in §26.8(i). …
##### 6: 26.7 Set Partitions: Bell Numbers
For $S\left(n,k\right)$ see §26.8(i). …
##### 7: 26.14 Permutations: Order Notation
In this subsection $S\left(n,k\right)$ is again the Stirling number of the second kind26.8), and $B_{m}$ is the $m$th Bernoulli number24.2(i)). …
26.14.7 $\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{j=0}^{n-k}(-1)^{n-k-j}j!\genfrac{(}{)}{0.0% pt}{}{n-j}{k}S\left(n,j\right),$
26.14.12 $S\left(n,m\right)=\frac{1}{m!}\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}% \genfrac{(}{)}{0.0pt}{}{k}{n-m},$ $n\geq m$, $n\geq 1$.
##### 8: 26.13 Permutations: Cycle Notation
The Stirling cycle numbers of the first kind, denoted by $\left[n\atop k\right]$, count the number of permutations of $\{1,2,\ldots,n\}$ with exactly $k$ cycles. They are related to Stirling numbers of the first kind by
##### 9: 24.16 Generalizations
24.16.8 $\beta_{n}(\lambda)=n!b_{n}\lambda^{n}+\sum_{k=1}^{\left\lfloor\ifrac{n}{2}% \right\rfloor}\frac{n}{2k}B_{2k}s\left(n-1,2k-1\right)\lambda^{n-2k},$ $n=2,3,\dots$.
Here $s\left(n,m\right)$ again denotes the Stirling number of the first kind. …
##### 10: 26.15 Permutations: Matrix Notation
26.15.13 $r_{n-k}(B)=S\left(n,k\right).$