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. Explicitly, the unsigned Lah numbers $L(n,k)$ are given by the formula involving the binomial coefficient

$L(n,k)={n-1 \choose k-1}{\frac {n!}{k!))$ for $n\geq k\geq 1$ .

Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a set of ${\textstyle n}$ elements can be partitioned into ${\textstyle k}$ nonempty linearly ordered subsets. Lah numbers are related to Stirling numbers.

For ${\textstyle n\geq 1}$ , the Lah number ${\textstyle L(n,1)}$ is equal to the factorial ${\textstyle n!}$ in the interpretation above, the only partition of ${\textstyle \{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)\},\{(3,2,1)\)$ ${\textstyle L(3,2)}$ is equal to 6, because there are six partitions of ${\textstyle \{1,2,3\))$ into two ordered parts:
$\{1,(2,3)\},\{1,(3,2)\},\{2,(1,3)\},\{2,(3,1)\},\{3,(1,2)\},\{3,(2,1)\)$ ${\textstyle L(n,n)}$ is always 1 because the only way to partition ${\textstyle \{1,2,\ldots ,n\))$ into $n$ non-empty subsets results in subsets of size 1, that can only be permuted in one way. In the more recent litterature, KaramataKnuth style notation has taken over. Lah numbers are now often written as
$L(n,k)=\left\lfloor {n \atop k}\right\rfloor$ ## Table of values

Below is a table of values for the Lah numbers:

k
n
0 1 2 3 4 5 6 7 8 9 10
0 1
1 0 1
2 0 2 1
3 0 6 6 1
4 0 24 36 12 1
5 0 120 240 120 20 1
6 0 720 1800 1200 300 30 1
7 0 5040 15120 12600 4200 630 42 1
8 0 40320 141120 141120 58800 11760 1176 56 1
9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1
10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1

The row sums are ${\textstyle 1,1,3,13,73,501,4051,37633,\dots }$ (sequence A000262 in the OEIS).

## Rising and falling factorials

Let ${\textstyle x^{(n)))$ represent the rising factorial ${\textstyle x(x+1)(x+2)\cdots (x+n-1)}$ and let ${\textstyle (x)_{n))$ represent the falling factorial ${\textstyle x(x-1)(x-2)\cdots (x-n+1)}$ . The Lah numbers are the coefficients that express each of these families of polynomials in terms of the other. Explicitly,

$x^{(n)}=\sum _{k=0}^{n}L(n,k)(x)_{k)$ and
$(x)_{n}=\sum _{k=0}^{n}(-1)^{n-k}L(n,k)x^{(k)}.$ For example,
$x(x+1)(x+2)={\color {red}6}x+{\color {red}6}x(x-1)+{\color {red}1}x(x-1)(x-2)$ and
$x(x-1)(x-2)={\color {red}6}x-{\color {red}6}x(x+1)+{\color {red}1}x(x+1)(x+2),$ 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 _{j=k}^{n}\left[{n \atop j}\right]\left$$(j \atop k}\right$$$ where ${\textstyle \left[{n \atop j}\right]}$ are the Stirling numbers of the first kind and ${\textstyle \left$$(j \atop k}\right$$)$ are the Stirling numbers of the second kind.

$L(n,k)={n-1 \choose k-1}{\frac {n!}{k!))={n \choose k}{\frac {(n-1)!}{(k-1)!))={n \choose k}{n-1 \choose k-1}(n-k)!$ $L(n,k)={\frac {n!(n-1)!}{k!(k-1)!))\cdot {\frac {1}{(n-k)!))=\left({\frac {n!}{k!))\right)^{2}{\frac {k}{n(n-k)!))$ $k(k+1)L(n,k+1)=(n-k)L(n,k)$ , for $k>0$ .

### Recurrence relations

The Lah numbers satisfy the recurrence relations

{\begin{aligned}L(n+1,k)&=(n+k)L(n,k)+L(n,k-1)\\&=k(k+1)L(n,k+1)+2kL(n,k)+L(n,k-1)\end{aligned)) where $L(n,0)=\delta _{n)$ , the Kronecker delta, and $L(n,k)=0$ for all $k>n$ .

### Exponential generating function

$\sum _{n\geq k}L(n,k){\frac {x^{n)){n!))={\frac {1}{k!))\left({\frac {x}{1-x))\right)^{k)$ ### Ordinary generating function

$\sum _{n\geq 0}L(n,k)x^{n}=x\prod _{k=1}^{\infty }{\frac {x+k}{1-kx))$ ### Derivative of exp(1/x)

The n-th derivative of the function $e^{\frac {1}{x))$ can be expressed with the Lah numbers, as follows

${\frac ((\textrm {d))^{n))((\textrm {d))x^{n))}e^{\frac {1}{x))=(-1)^{n}\sum _{k=1}^{n}{\frac {L(n,k)}{x^{n+k))}\cdot e^{\frac {1}{x)).$ For example,

${\frac {\textrm {d))((\textrm {d))x))e^{\frac {1}{x))=-{\frac {1}{x^{2))}\cdot e^{\frac {1}{x))$ ${\frac ((\textrm {d))^{2))((\textrm {d))x^{2))}e^{\frac {1}{x))={\frac {\textrm {d))((\textrm {d))x))\left(-{\frac {1}{x^{2))}e^{\frac {1}{x))\right)=-{\frac {-2}{x^{3))}\cdot e^{\frac {1}{x))-{\frac {1}{x^{2))}\cdot {\frac {-1}{x^{2))}\cdot e^{\frac {1}{x))=\left({\frac {2}{x^{3))}+{\frac {1}{x^{4))}\right)\cdot e^{\frac {1}{x))$ ${\frac ((\textrm {d))^{3))((\textrm {d))x^{3))}e^{\frac {1}{x))={\frac {\textrm {d))((\textrm {d))x))\left(\left({\frac {2}{x^{3))}+{\frac {1}{x^{4))}\right)\cdot e^{\frac {1}{x))\right)=\left({\frac {-6}{x^{4))}+{\frac {-4}{x^{5))}\right)\cdot e^{\frac {1}{x))+\left({\frac {2}{x^{3))}+{\frac {1}{x^{4))}\right)\cdot {\frac {-1}{x^{2))}\cdot e^{\frac {1}{x))=-\left({\frac {6}{x^{4))}+{\frac {6}{x^{5))}+{\frac {1}{x^{6))}\right)\cdot e^{\frac {1}{x))$ Generalized Laguerre polynomials $L_{n}^{(\alpha )}(x)$ are linked to Lah numbers upon setting $\alpha =-1$ $n!L_{n}^{(-1)}(x)=\sum _{k=0}^{n}L(n,k)(-x)^{k)$ This formula is the default Laguerre polynomial in Umbral calculus convention.

## 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—$O(n\log n)$ —of their integer coefficients. The Lah and Laguerre transforms naturally arise in the perturbative description of the chromatic dispersion. In Lah-Laguerre optics, such an approach tremendously speeds up optimization problems.

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3. ^ Petkovsek, Marko; Pisanski, Tomaz (Fall 2007). "Combinatorial Interpretation of Unsigned Stirling and Lah Numbers". Pi Mu Epsilon Journal. 12 (7): 417–424. JSTOR 24340704.
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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. 338 (10): 1660–1666. doi:10.1016/j.disc.2014.03.029. ISSN 0012-365X.
7. ^ Daboul, Siad; Mangaldan, Jan; Spivey, Michael Z.; Taylor, Peter J. (2013). "The Lah Numbers and the nth Derivative of $e^{1 \over x)$ ". Mathematics Magazine. 86 (1): 39–47. doi:10.4169/math.mag.86.1.039. JSTOR 10.4169/math.mag.86.1.039. S2CID 123113404.
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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). doi:10.1002/ett.3984. S2CID 225866797.
10. ^ "Image Steganography-using-Lah-Transform". MathWorks.
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. Bibcode:2022OExpr..3040779P. doi:10.1364/OE.457139. PMID 36299007.
12. ^ Popmintchev, Dimitar; Wang, Siyang; Xiaoshi, Zhang; Stoev, Ventzislav; Popmintchev, Tenio (2020-08-30). "Theory of the Chromatic Dispersion, Revisited". arXiv:2011.00066 [physics.optics].
• The signed and unsigned Lah numbers are respectively (sequence A008297 in the OEIS) and (sequence A105278 in the OEIS)