§26.1 Special Notation

Keywords:: Stirling numbers (first and second kinds), combinatorics, partitions
Permalink:: http://dlmf.nist.gov/26.1

(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 $A$ .
$j\divides k$	$j$ divides $k$ .
$(h,k)$	greatest common divisor of positive integers $h$ and $k$ .

The main functions treated in this chapter are:

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

¶ Alternative Notations

Keywords:: factorials (rising or falling)

Many combinatorics references use the rising and falling factorials:

26.1.1

${x}^{{\overline{n}}}=x(x+1)(x+2)\cdots(x+n-1),$

${x}^{{\underline{n}}}=x(x-1)(x-2)\cdots(x-n+1).$

Symbols:: $x$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.1.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

Other notations for $\mathop{s\/}\nolimits\!\left(n,k\right)$ , 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)), $\binom{n-1}{k-1}B_{{n-k}}^{{(n)}}$ (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 $\mathop{S\/}\nolimits\!\left(n,k\right)$ , 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)), $\binom{n}{k}B_{{n-k}}^{{(-k)}}$ (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).