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

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov