Editing Lah number (section)

{{Short description|Mathematical sequence}}
{{Use American English|date = March 2019}}
[[File:Lah numbers.svg|thumb|upright=1.35|Illustration of the unsigned Lah numbers for ''n'' and ''k'' between 1 and 4]]
In [[mathematics]], the '''(signed and unsigned) Lah numbers''' are [[coefficient]]s expressing [[rising factorial]]s in terms of [[falling factorial]]s and vice versa.  They were discovered by [[Ivo Lah]] in 1954.<ref>{{cite journal | first=Ivo | last=Lah | title=A new kind of numbers and its application in the actuarial mathematics | journal=Boletim do Instituto dos Actuários Portugueses | volume=9 | year=1954 | pages=7–15}}</ref><ref>[https://books.google.com/books?id=zWgIPlds29UC John Riordan, ''Introduction to Combinatorial Analysis''], Princeton University Press (1958, reissue 1980) {{isbn|978-0-691-02365-6}} (reprinted again in 2002 by Dover Publications).</ref>  Explicitly, the unsigned Lah numbers <math>L(n, k)</math> are given by the formula involving the [[binomial coefficient]] 

<math display="block"> L(n,k) = {n-1 \choose k-1} \frac{n!}{k!}</math>

for <math>n \geq k \geq 1</math>.

Unsigned Lah numbers have an interesting meaning in [[combinatorics]]: they count the number of ways a [[Set (mathematics)|set]] of ''<math display="inline">n</math>'' elements can be [[Partition of a set|partition]]ed into ''<math display="inline">k</math>'' nonempty linearly ordered [[subset]]s.<ref>{{cite journal | title=Combinatorial Interpretation of Unsigned Stirling and Lah Numbers | first1=Marko | last1=Petkovsek | first2=Tomaz | last2=Pisanski | journal=Pi Mu Epsilon Journal | volume=12 | number=7 | date = Fall 2007 | pages=417–424 | jstor=24340704}}</ref> Lah numbers are related to [[Stirling number]]s.<ref>{{cite book | first=Louis | last=Comtet | title=Advanced Combinatorics | publisher=Reidel | location=Dordrecht, Holland | year=1974 | url=https://archive.org/details/Comtet_Louis_-_Advanced_Coatorics | page=[https://archive.org/details/Comtet_Louis_-_Advanced_Coatorics/page/n83 156]| isbn=9789027703804 }}</ref>

For <math display="inline">n \geq 1</math>, the Lah number <math display="inline">L(n, 1)</math> is equal to the [[factorial]] <math display="inline">n!</math> in the interpretation above, the only partition of <math display="inline">\{1, 2, 3 \}</math> into 1 set can have its set ordered in 6 ways:<math display="block">\{(1, 2, 3)\}, \{(1, 3, 2)\}, \{(2, 1, 3)\}, \{(2, 3, 1)\}, \{(3, 1, 2)\}, \{(3, 2, 1)\}</math><math display="inline">L(3, 2)</math> is equal to 6, because there are six partitions of <math display="inline">\{1, 2, 3 \}</math> into two ordered parts:<math display="block">\{1, (2, 3) \}, \{1, (3, 2) \}, \{2, (1, 3) \}, \{2, (3, 1) \}, \{3, (1, 2) \}, \{3, (2, 1) \}</math><math display="inline">L(n, n)</math> is always 1 because the only way to partition <math display="inline">\{1, 2, \ldots, n\}</math> into <math>n</math> non-empty subsets results in subsets of size 1, that can only be permuted in one way.
In the more recent literature,<ref>{{cite arXiv |eprint=1412.8721 |first=Mark |last=Shattuck |title=Generalized r-Lah numbers |date=2014|class=math.CO }}</ref><ref>{{Cite journal |last1=Nyul |first1=Gábor |last2=Rácz |first2=Gabriella |date=2015-10-06 |title=The r-Lah numbers |url=https://www.sciencedirect.com/science/article/pii/S0012365X14001241 |journal=Discrete Mathematics |series=Seventh Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications, Košice 2013 |language=en |volume=338 |issue=10 |pages=1660–1666 |doi=10.1016/j.disc.2014.03.029 |issn=0012-365X|hdl=2437/213886 |hdl-access=free }}</ref> [[Jovan Karamata|Karamata]]–[[Donald Knuth|Knuth]] style notation has taken over. Lah numbers are now often written as<math display="block">L(n,k) = \left\lfloor {n \atop k} \right\rfloor</math>