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 1955,^[1] are coefficients expressing rising factorials in terms of falling factorials.

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 OEIS):

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

Signed Lah numbers (sequence A008297 in 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:

$L(n,k)=\left\lfloor\begin{matrix} n \\ k \end{matrix}\right\rfloor.$

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

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

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

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

$L(n,k) = \sum_{j} \left[{n\atop j}\right] \left\{{j\atop k}\right\},$ 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, and with the conventions $L(0,0)=1$ and $L(n , k )=0$ if $k>n$.

$L(n,1) = n!$

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

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

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

$L(n,n) = 1$

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

Table of values

Below is a table of values for the Lah numbers:

$_n\!\!\diagdown\!\!^k$	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

References

1. 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).