Lah number

Illustration of the unsigned Lah numbers for n and k between 1 and 4

In mathematics, the Lah numbers, discovered by Ivo Lah in 1954, are coefficients expressing rising factorials in terms of falling factorials. They are also the coefficients of the $n$th derivatives of $e^{1 \over x}$.

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. Lah numbers are related to Stirling numbers.

Unsigned Lah numbers (sequence A105278 in the OEIS):

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

Signed Lah numbers (sequence A008297 in the OEIS):

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

L(n, 1) is always n!; in 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 since, e.g., partitioning {1, 2, 3} into 3 non-empty subsets results in subsets of length 1.

{(1), (2), (3)}

Adapting the Karamata–Knuth notation for Stirling numbers, it has been proposed to use the following alternative notation for Lah numbers:

$$L(n,k)=\sum _{j=0}^{n}\left[{\begin{matrix}n\\j\end{matrix}}\right]\left\{{\begin{matrix}j\\k\end{matrix}}\right\}\,$$

Rising and falling factorials

Let $x^{(n)}$ represent the rising factorial $x(x+1)(x+2)\cdots (x+n-1)$ and let $(x)_{n}$ represent the falling factorial $x(x-1)(x-2)\cdots (x-n+1)$.

Then $x^{(n)}=\sum _{k=1}^{n}L(n,k)(x)_{k}$ and $(x)_{n}=\sum _{k=1}^{n}(-1)^{n-k}L(n,k)x^{(k)}.$

For example, $$x(x+1)(x+2)={\color {red}6}x+{\color {red}6}x(x-1)+{\color {red}1}x(x-1)(x-2).$$

Compare the third row of the table of values.

Identities and relations

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

$L(n,k)={\frac {n!(n-1)!}{k!(k-1)!}}\cdot {\frac {1}{(n-k)!}}=\left({\frac {n!}{k!}}\right)^{2}{\frac {k}{n(n-k)!}}$

$L(n,k+1)={\frac {n-k}{k(k+1)}}L(n,k).$

$L(n+1,k)=(n+k)L(n,k)+L(n,k-1)$ where $L(n,0)=0$, $L(n,k)=0$ for all $k>n$, and $L(1,1)=1.$

$L(n,k)=\sum _{j}\left[{n \atop j}\right]\left\{{j \atop k}\right\},$ where $\left[{n \atop j}\right]$ are the Stirling numbers of the first kind and $\left\{{j \atop k}\right\}$ are the Stirling numbers of the second kind, $L(0,0)=1$, and $L(n,k)=0$ for all $k>n$.

$L(n,1)=n!$

$L(n,2)=(n-1)n!/2$

$L(n,3)=(n-2)(n-1)n!/12$

$L(n,n-1)=n(n-1)$

$L(n,n)=1$

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

Table of values

Below is a table of values for the Lah numbers:

k n	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 numbers
• Pascal matrix