Editing Stirling number (section)

==Notation==
{{main|Stirling numbers of the first kind|Stirling numbers of the second kind}}
Several different notations for Stirling numbers are in use. Ordinary (signed) '''Stirling numbers of the first kind''' are commonly denoted:

: <math> s(n,k)\,.</math>
'''Unsigned Stirling numbers of the first kind''', which count the number of [[permutation]]s of ''n'' elements with ''k'' disjoint [[cyclic permutation|cycle]]s, are denoted:

: <math> \biggl[{n \atop k}\biggr] =c(n,k)=|s(n,k)|=(-1)^{n-k} s(n,k)\,</math>
'''Stirling numbers of the second kind''', which count the number of ways to partition a set of ''n'' elements into ''k'' nonempty subsets:<ref>Ronald L. Graham, Donald E. Knuth, Oren Patashnik (1988) ''[[Concrete Mathematics]]'', Addison-Wesley, Reading MA. {{isbn|0-201-14236-8}}, p.&nbsp;244.</ref>

: <math> \biggl\{{\!n\! \atop \!k\!}\biggr\} = S(n,k) = S_n^{(k)} \,</math>

[[Abramowitz and Stegun]] use an uppercase <math>S</math> and a [[blackletter]] <math>\mathfrak S</math>, respectively, for the first and second kinds of Stirling number.  The notation of brackets and braces, in analogy to [[binomial coefficients]], was introduced in 1935 by [[Jovan Karamata]] and promoted later by [[Donald Knuth]], though the bracket notation conflicts with a common notation for [[Gaussian coefficient]]s.<ref>{{Cite journal |last=Knuth |first=Donald E. |date=1992 |title=Two Notes on Notation |url=https://www.jstor.org/stable/2325085 |journal=American Mathematical Monthly |volume=99 |issue=5 |pages=403–422 |doi=10.2307/2325085 |jstor=2325085 }}</ref> The mathematical motivation for this type of notation, as well as additional Stirling number formulae, may be found on the page for [[Stirling numbers and exponential generating functions]].

Another infrequent notation is <math>s_1(n,k)</math> and <math>s_2(n,k)</math>.

==Expansions of falling and rising factorials<span class="anchor" id="falling_and_rising_anchor"></span>==
Stirling numbers express coefficients in expansions of [[falling and rising factorials]] (also known as the Pochhammer symbol) as polynomials.

That is, the '''falling factorial''', defined as <math>\ (x)_{n} = x(x-1)\ \cdots(x-n+1)\ ,</math> is a polynomial in {{mvar|x}} of degree {{mvar|n}} whose expansion is

:<math>(x)_{n}\ =\ \sum_{k=0}^n\ s(n,k)\ x^k\ </math>

with (signed) Stirling numbers of the first kind as coefficients.

Note that <math>\ (x)_0 \equiv 1\ ,</math> by convention, because it is an [[empty product]]. The notations <math style="vertical-align:baseline;">\ x^{\underline{n}}\ </math> for the falling factorial and <math style="vertical-align:baseline;">\ x^{\overline{n}}\ </math> for the rising factorial are also often used.<ref>{{cite book |last=Aigner |first=Martin |title=A Course in Enumeration |url=https://archive.org/details/courseenumeratio00aign_007 |url-access=limited |publisher=Springer |year=2007 |pages=[https://archive.org/details/courseenumeratio00aign_007/page/n563 561] |chapter=Section 1.2 – Subsets and binomial coefficients |isbn=978-3-540-39032-9}}</ref> (Confusingly, the Pochhammer symbol that many use for ''falling'' factorials is used in [[special function]]s for ''rising'' factorials.)

Similarly, the '''rising factorial''', defined as <math>\ x^{(n)}\ =\ x(x+1)\ \cdots(x+n-1)\ ,</math> is a polynomial in {{mvar|x}} of degree {{mvar|n}} whose expansion is

:<math> x^{(n)}\ =\ \sum_{k=0}^n\ \biggl[{n \atop k}\biggr]\ x^k\ =\ \sum_{k=0}^n\ (-1)^{n-k}\ s(n,k)\ x^k\ ,</math>

with unsigned Stirling numbers of the first kind as coefficients.
One of these expansions can be derived from the other by observing that <math>\ x^{(n)} = (-1)^n (-x)_{n} ~.</math> 

Stirling numbers of the second kind express the reverse relations:

:<math>\ x^n\ =\ \sum_{k=0}^n\ S(n,k)\ (x)_k\ </math>

and

:<math>\ x^n\ =\ \sum_{k=0}^n\ (-1)^{n-k}\ S(n,k)\ x^{(k)} ~.</math>