26 Combinatorial Analysis Notation 26 Combinatorial Analysis 26.2 Basic Definitions

§26.1 Special Notation

ⓘ

Defines:: $\left|\NVar{A}\right|$ : number of elements of a finite set $A$
Keywords:: Stirling numbers (first and second kinds), combinatorics, notations, partitions, plane, restricted
Permalink:: http://dlmf.nist.gov/26.1
See also:: Annotations for Ch.26

(For other notation see Notation for the Special Functions.)

$x$	real variable.
$h,j,k,\ell,m,n$	nonnegative integers.
$\lambda$	integer partition.
$\pi$	plane partition.
$\left\|A\right\|$	number of elements of a finite set $A$ .
$j\mathbin{\|}k$	$j$ divides $k$ .
$\left(h,k\right)$	greatest common divisor of positive integers $h$ and $k$ .

The main functions treated in this chapter are:

$\genfrac{(}{)}{0.0pt}{}{m}{n}$	binomial coefficient.
$\genfrac{(}{)}{0.0pt}{}{m}{n_{1},n_{2},\ldots,n_{k}}$	multinomial coefficient.
$\genfrac{<}{>}{0.0pt}{}{m}{n}$	Eulerian number.
$\genfrac{[}{]}{0.0pt}{}{m}{n}_{q}$	Gaussian polynomial.
$B\left(n\right)$	Bell number.
$C\left(n\right)$	Catalan number.
$p\left(n\right)$	number of partitions of $n$ .
$p_{k}\left(n\right)$	number of partitions of $n$ into at most $k$ parts.
$\operatorname{pp}\left(n\right)$	number of plane partitions of $n$ .
$s\left(n,k\right)$	Stirling numbers of the first kind.
$S\left(n,k\right)$	Stirling numbers of the second kind.

Alternative Notations

ⓘ

Keywords:: factorials (rising or falling)
See also:: Annotations for §26.1 and Ch.26

Many combinatorics references use the rising and falling factorials:

26.1.1	$\displaystyle{x}^{\overline{n}}$	$\displaystyle=x(x+1)(x+2)\cdots(x+n-1),$
26.1.1	$\displaystyle{x}^{\underline{n}}$	$\displaystyle=x(x-1)(x-2)\cdots(x-n+1).$
ⓘ Symbols: $x$ : real variable and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.1.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §26.1, §26.1 and Ch.26

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).

26 Combinatorial Analysis 26.2 Basic Definitions

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