Lah number explained

In mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were discovered by Ivo Lah in 1954.[1] [2] Explicitly, the unsigned Lah numbers

L(n,k) = \frac

for

n\geqk\geq1

.

Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets.[3] Lah numbers are related to Stirling numbers.[4]

For n \geq 1, the Lah number L(n, 1) is equal to the factorial n! in the interpretation above, the only partition of \ into 1 set can have its set ordered in 6 ways:\, \, \, \, \, \L(3, 2) is equal to 6, because there are six partitions of \ into two ordered parts:\, \, \, \, \, \L(n, n) is always 1 because the only way to partition \ into

n

non-empty subsets results in subsets of size 1, that can only be permuted in one way.In the more recent literature,[5] [6] KaramataKnuth style notation has taken over. Lah numbers are now often written asL(n,k) = \left\lfloor \right\rfloor

Table of values

Below is a table of values for the Lah numbers:

01 2 3 4 5 6 7 8 9 10
01
101
2021
30661
402436121
50120240120201
6072018001200300301
70504015120126004200630421
804032014112014112058800117601176561
9036288014515201693440846720211680282242016721
10036288001632960021772800127008003810240635040604803240901

The row sums are 1, 1, 3, 13, 73, 501, 4051, 37633, \dots .

Rising and falling factorials

Let x^ represent the rising factorial x(x+1)(x+2) \cdots (x+n-1) and let (x)_n represent the falling factorial x(x-1)(x-2) \cdots (x-n+1). The Lah numbers are the coefficients that express each of these families of polynomials in terms of the other. Explicitly,x^ = \sum_^n L(n,k) (x)_kand(x)_n = \sum_^n (-1)^ L(n,k)x^.For example,x(x+1)(x+2) = x + x(x-1) + x(x-1)(x-2)andx(x-1)(x-2) = x - x(x+1) + x(x+1)(x+2),

where the coefficients 6, 6, and 1 are exactly the Lah numbers

L(3,1)

,

L(3,2)

, and

L(3,3)

.

Identities and relations

The Lah numbers satisfy a variety of identities and relations.

In KaramataKnuth notation for Stirling numbers L(n,k) = \sum_^n \left[{n\atop j}\right] \left\where \left[{n\atop j}\right] are the Stirling numbers of the first kind and \left\ are the Stirling numbers of the second kind.

L(n,k)={n-1\choosek-1}

n!
k!

={n\choosek}

(n-1)!
(k-1)!

={n\choosek}{n-1\choosek-1}(n-k)!

L(n,k)=

n!(n-1)!
k!(k-1)!
1
(n-k)!

=\left(

n!
k!

\right

2k
n(n-k)!
)

k(k+1)L(n,k+1)=(n-k)L(n,k)

, for

k>0

.

Recurrence relations

The Lah numbers satisfy the recurrence relations\begin L(n+1,k) &= (n+k) L(n,k) + L(n,k-1) \\&= k(k+1) L(n, k+1) + 2k L(n, k) + L(n, k-1)\endwhere

L(n,0)=\deltan

, the Kronecker delta, and

L(n,k)=0

for all

k>n

.

Exponential generating function

\sumn\geqL(n,k)

xn
n!

=

1
k!

\left(

x
1-x

\right)k

Derivative of exp(1/x)

The n-th derivative of the function

1{x}
e
can be expressed with the Lah numbers, as follows[7] \frac e^\frac1x = (-1)^n \sum_^n \frac \cdot e^\frac1x.For example,
rmd
rmdx
1x
=
e

-

1
x2

1x
e
rmd2
rmdx2
1{x}
=
e
rmd\left(-
rmdx
1{x
2}
1x
e

\right)=-

-2
x3

1x
e

-

1{x
2}

-1
x2

1x
e

=\left(

2{x
3}

+

1{x
4}\right)

1x
e
rmd3
rmdx3
1{x}
=
e
rmd
rmdx

\left(\left(

2{x
3}

+

1{x
4}\right)

1x
e

\right)=\left(

-6
x4

+

-4
x5

\right)

1x
e

+\left(

2{x
3}

+

1{x
4}\right)

-1
x2

1x
e=-\left(
6{x
4}

+

6{x
5}

+

1{x
6}\right)

1
x
e

Link to Laguerre polynomials

(\alpha)
L
n(x)
are linked to Lah numbers upon setting

\alpha=-1

n! L_n^(x) =\sum_^n L(n,k) (-x)^kThis formula is the default Laguerre polynomial in Umbral calculus convention.[8]

Practical application

In recent years, Lah numbers have been used in steganography for hiding data in images. Compared to alternatives such as DCT, DFT and DWT, it has lower complexity of calculation -

O(nlogn)

- of their integer coefficients.[9] [10] The Lah and Laguerre transforms naturally arise in the perturbative description of the chromatic dispersion.[11] [12] In Lah-Laguerre optics, such an approach tremendously speeds up optimization problems.

See also

References

  1. Ivo . Lah . A new kind of numbers and its application in the actuarial mathematics . Boletim do Instituto dos Actuários Portugueses . 9 . 1954 . 7–15.
  2. https://books.google.com/books?id=zWgIPlds29UC John Riordan, Introduction to Combinatorial Analysis
  3. Combinatorial Interpretation of Unsigned Stirling and Lah Numbers . Marko . Petkovsek . Tomaz . Pisanski . Pi Mu Epsilon Journal . 12 . 7 . Fall 2007 . 417–424 . 24340704.
  4. Book: Comtet, Louis . Advanced Combinatorics . Reidel . Dordrecht, Holland . 1974 . 156. 9789027703804 .
  5. 1412.8721 . Mark . Shattuck . Generalized r-Lah numbers . 2014. math.CO .
  6. Nyul . Gábor . Rácz . Gabriella . 2015-10-06 . The r-Lah numbers . Discrete Mathematics . Seventh Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications, Košice 2013 . en . 338 . 10 . 1660–1666 . 10.1016/j.disc.2014.03.029 . 0012-365X. 2437/213886 . free .
  7. Daboul . Siad . Mangaldan . Jan . Spivey . Michael Z. . Taylor . Peter J. . 2013 . The Lah Numbers and the nth Derivative of

    e1\over

    . Mathematics Magazine . 86 . 39–47 . 10.4169/math.mag.86.1.039 . 10.4169/math.mag.86.1.039 . 123113404 . 1.
  8. Rota . Gian-Carlo . Kahaner . D . Odlyzko . A . 1973-06-01 . On the foundations of combinatorial theory. VIII. Finite operator calculus . Journal of Mathematical Analysis and Applications . en . 42 . 3 . 684–760 . 10.1016/0022-247X(73)90172-8 . 0022-247X. free .
  9. Ghosal . Sudipta Kr . Mukhopadhyay . Souradeep . Hossain . Sabbir . Sarkar . Ram . 2020 . Application of Lah Transform for Security and Privacy of Data through Information Hiding in Telecommunication . Transactions on Emerging Telecommunications Technologies . 32 . 2 . 10.1002/ett.3984. 225866797 .
  10. Web site: Image Steganography-using-Lah-Transform . MathWorks. 5 June 2020 .
  11. Popmintchev. Dimitar. Wang. Siyang. Xiaoshi . Zhang . Stoev. Ventzislav. Popmintchev. Tenio. 2022-10-24 . Analytical Lah-Laguerre optical formalism for perturbative chromatic dispersion. Optics Express. 30. 22. 40779–40808. 10.1364/OE.457139. free. 36299007 . 2022OExpr..3040779P.
  12. Popmintchev. Dimitar. Wang. Siyang. Xiaoshi. Zhang . Stoev. Ventzislav. Popmintchev. Tenio. 2020-08-30. Theory of the Chromatic Dispersion, Revisited . physics.optics . 2011.00066.

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