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In [[mathematics]], '''Lah numbers''', discovered by [[Ivo Lah]] in 1955,<ref>[http://books.google.com/books?id=zWgIPlds29UC ''Introduction to Combinatorial Analysis''] Princeton University Press (1958, reissue 1980) ISBN 978-0-691-02365-6 (reprinted again in 2002, by Courier Dover Publications).</ref> are [[coefficient]]s expressing [[rising factorial]]s in terms of [[falling factorial]]s.
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'''Unsigned Lah numbers''' have an interesting meaning in [[combinatorics]]: they count the number of ways a [[Set (mathematics)|set]] of ''n'' elements can be [[Partition of a set|partition]]ed into ''k'' nonempty linearly ordered [[subset]]s. Lah numbers are related to [[Stirling number]]s.
 
Unsigned Lah numbers:
 
:<math> L(n,k) = {n-1 \choose k-1} \frac{n!}{k!}.</math>
 
Signed Lah numbers:
 
:<math> L'(n,k) = (-1)^n {n-1 \choose k-1} \frac{n!}{k!}.</math>
 
''L''(''n'', 1) is always ''n''!; using the interpretation above, the only partition of {1, 2, 3} into 1 set can have its set ordered in 6 ways:
 
:{(1, 2, 3)}, {(1, 3, 2)}, {(2, 1, 3)}, {(2, 3, 1)}, {(3, 1, 2)} or {(3, 2, 1)}
 
''L''(3, 2) corresponds to the 6 partitions with two ordered parts:
 
:{(1), (2, 3)}, {(1), (3, 2)}, {(2), (1, 3)}, {(2), (3, 1)}, {(3), (1, 2)} or {(3), (2, 1)}
 
''L''(''n'', ''n'') is always 1; e.g., partitioning {1, 2, 3} into 3 non-empty subsets results in subsets of length 1.
 
:{(1), (2), (3)}
 
Paraphrasing Karamata-Knuth notation for [[Stirling numbers]], it was
proposed to use the following alternative notation for Lah numbers:
 
:<math>L(n,k)=\left\lfloor\begin{matrix} n \\ k \end{matrix}\right\rfloor.</math>
 
==Rising and falling factorials==
 
Let <math>x^{(n)}</math> represent the rising factorial <math>x(x+1)(x+2) \cdots (x+n-1)</math> and let <math>(x)_n</math> represent the falling factorial <math>x(x-1)(x-2) \cdots (x-n+1)</math>.
 
Then <math>x^{(n)} = \sum_{k=1}^n L(n,k) (x)_k</math> and <math>(x)_n = \sum_{k=1}^n (-1)^{n-k} L(n,k)x^{(k)}.</math>
 
For example, <math>x(x+1)(x+2) = {\color{Red}6}x + {\color{Red}6}x(x-1) + {\color{Red}1}x(x-1)(x-2).</math>
 
Compare the third row of the table of values.
 
==Identities and relations==
 
:<math> L(n,k) = {n-1 \choose k-1} \frac{n!}{k!} = {n \choose k} \frac{(n-1)!}{(k-1)!}</math>
:<math> L(n,k) = \frac{n!(n-1)!}{k!(k-1)!}\cdot\frac{1}{(n-k)!} = \left (\frac{n!}{k!} \right )^2\frac{k}{n(n-k)!}</math>
:<math> L(n,k+1) = \frac{n-k}{k(k+1)} L(n,k).</math>
:<math> L(n,k) = \sum_{j} \left[{n\atop j}\right] \left\{{j\atop k}\right\},</math> with <math>\left[{n\atop j}\right]</math> the Stirling numbers of the first kind, <math>\left\{{j\atop k}\right\}</math> the Stirling numbers of the second kind and with the conventions <math>L(0,0)=1</math> and <math>L(n , k )=0</math> if <math>k>n</math>.
 
:<math> L(n,1) = n!</math>
:<math> L(n,2) = (n-1)n!/2</math>
:<math> L(n,3) = (n-2)(n-1)n!/12</math>
:<math> L(n,n-1) = n(n-1)</math>
:<math> L(n,n) = 1</math>
 
==Table of values==
Below is a table of values for the Lah numbers:
{| class="wikitable" style="text-align:right;"
|-
! <math>_n\!\!\diagdown\!\!^k</math> !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12
|-
! 1
| 1
|-
! 2
| 2
| 1
|-
! 3
| 6
| 6
| 1
|-
! 4
| 24
| 36
| 12
| 1
|-
! 5
| 120
| 240
| 120
| 20
| 1
|-
! 6
| 720
| 1800
| 1200
| 300
| 30
| 1
|-
! 7
| 5040
| 15120
| 12600
| 4200
| 630
| 42
| 1
|-
! 8
| 40320
| 141120
| 141120
| 58800
| 11760
| 1176
| 56
| 1
|-
! 9
| 362880
| 1451520
| 1693440
| 846720
| 211680
| 28224
| 2016
| 72
| 1
|-
! 10
|3628800
|16329600
|21772800
|12700800
|3810240
|635040
|60480
|3240
|90
|1
|-
! 11
|39916800
|199584000
|299376000
|199584000
|69854400
|13970880
|1663200
|11880
|4950
|110
|1
|-
! 12
|479001600
|2634508800
|4390848000
|3293136000
|1317254400
|307359360
|43908480
|3920400
|217800
|7260
|132
|1
|}
 
== See also ==
* [[Stirling number]]s
* [[Pascal matrix]]
 
==References==
<references />
 
{{DEFAULTSORT:Lah Number}}
[[Category:Factorial and binomial topics]]
[[Category:Integer sequences]]
[[Category:Triangles of numbers]]

Latest revision as of 15:07, 12 January 2015

If you have the desire to process settings swiftly, loading files fast, but a body is logy and torpid, what would we do? If you are a giant "switchboard" which is deficiency of efficient management program plus effective housekeeper, what would you do? If you have send your exact commands to the notice, but the body can not perform correctly, what would you do? Yes! You need a full-featured repair registry!

Many of the reliable businesses usually offer a full cash back guarantee. This means that we have the chance to get your cash back in the event you find the registry cleaning has not delivered what you expected.

The Windows registry is a system database of info. Windows and alternative software store a great deal of settings plus additional info inside it, and retrieve such info from the registry all time. The registry is also a bottleneck in that because it is very the heart of the operating program, any issues with it could cause errors and bring the operating system down.

If you feel we don't have enough cash at the time to upgrade, then the number one choice is to free up several area by deleting several of the unwelcome files and folders.

One other option when arresting the 1328 error is to clean out a PC's registry. The registry is important as it happens to be where settings plus files selected by Windows for running are stored. As it really is frequently selected, breakdowns plus instances of files getting corrupted are not uncommon. Also as a result of the technique it is very configured, the "registry" gets saved inside the incorrect fashion constantly, which makes the program run slow, eventually causing your PC to suffer from a series of errors. The most effective method 1 may use inside cleaning out registries is to employ a reliable zookaware system. A registry cleaner will find out plus repair corrupted registry files plus settings allowing one's computer to run usually again.

Another factor is registry. It is one of the most crucial piece inside a Windows XP, Vista running systems. When Windows start, it read connected information from registry plus load into computer RAM. This takes up a big piece of the startup time. After the information is all loaded, computer runs the startup programs.

In different words, when the PC has any corrupt settings inside the registry database, these settings usually make the computer run slower plus with a lot of mistakes. And regrettably, it's the case which XP is prone to saving countless settings from the registry in the wrong technique, making them unable to run correctly, slowing it down plus causing a great deal of errors. Each time you employ your PC, it has to read 100's of registry settings... and there are usually thus many files open at when that XP gets confuse plus saves various in the incorrect means. Fixing these damaged settings might boost the speed of your program... plus to do which, we should look to use a 'registry cleaner'.

We can click here to obtain out how to accelerate Windows plus heighten PC perfomance. And you are able to click here to download a registry cleaner to aid we clean up registry.

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