Editing Stirling number (section)

==Lah numbers==
{{main|Lah numbers}}

The Lah numbers <math>L(n,k) = {n-1 \choose k-1} \frac{n!}{k!}</math> are sometimes called Stirling numbers of the third kind.<ref>{{cite book
 | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav
 | title=Handbook of Number Theory II  | publisher=[[Kluwer Academic Publishers]] | year=2004
 | url=https://books.google.com/books?id=B2WZkvmFKk8C&dq=%22Stirling+numbers+of+the+third+kind%22&pg=PA464 | isbn=9781402025464 | page=464}}</ref>
By convention, <math>L(0,0)=1</math> and <math>L(n,k)=0</math> if <math>n<k</math> or <math>k = 0 < n</math>.

These numbers are coefficients expressing falling factorials in terms of rising factorials and vice versa:
:<math>x^{(n)} = \sum_{k=0}^n L(n,k) (x)_k\quad</math> and <math>\quad(x)_n = \sum_{k=0}^n (-1)^{n-k} L(n,k)x^{(k)}.</math>

As above, this means they express the change of basis between the bases <math>(x)_0,(x)_1,(x)_2,\cdots</math> and <math>x^{(0)},x^{(1)},x^{(2)},\cdots</math>, completing the diagram.
In particular, one formula is the inverse of the other, thus:

: <math>\sum_{j=k}^n (-1)^{j-k} L(n,j) L(j,k) = \delta_{n,k}.</math>

Similarly, composing the change of basis from <math>x^{(n)}</math> to <math>x^n</math> with the change of basis from <math>x^n</math> to <math>(x)_{n}</math> gives the change of basis directly from <math>x^{(n)}</math> to <math>(x)_{n}</math>:

:<math> L(n,k) = \sum_{j=k}^n \biggl[{n \atop j}\biggr]  \biggl\{{\!j\! \atop \!k\!}\biggr\} ,</math>

and similarly for other compositions. In terms of matrices, if <math>L</math> denotes the matrix with entries <math>L_{nk}=L(n,k)</math> and <math>L^{-}</math> denotes the matrix with entries  <math>L^{-}_{nk}=(-1)^{n-k}L(n,k)</math>, then one is the inverse of the other: <math> L^{-} = L^{-1}</math>.
Composing the matrix of unsigned Stirling numbers of the first kind with the matrix of Stirling numbers of the second kind gives the Lah numbers: <math>L = |s| \cdot S</math>.

[[Enumerative combinatorics|Enumeratively]], <math display="inline">\left\{{\!n\! \atop \!k\!}\right\}, \left[{n \atop k}\right] , L(n,k)</math> can be defined as the number of partitions of ''n'' elements into ''k'' non-empty unlabeled subsets, where each subset is endowed with no order, a [[cyclic order]], or a linear order, respectively. In particular, this implies the inequalities:
: <math>\biggl\{{\!n\! \atop \!k\!}\biggr\} \leq \biggl[{n \atop k}\biggr] \leq L(n,k).</math>