§26.1 Special Notation

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

$x$ real variable. nonnegative integers. integer partition. plane partition. number of elements of a finite set $A$. $j$ divides $k$. greatest common divisor of positive integers $h$ and $k$.

The main functions treated in this chapter are:

$\binom{m}{n}$ binomial coefficient. multinomial coefficient. Eulerian number. Gaussian polynomial. Bell number. Catalan number. number of partitions of $n$. number of partitions of $n$ into at most $k$ parts. number of plane partitions of $n$. Stirling numbers of the first kind. Stirling numbers of the second kind.

Alternative Notations

Many combinatorics references use the rising and falling factorials:

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

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).