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In mathematics, 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:

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

Signed Lah numbers:

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

L(n, 1) is always n!; using the interpretation above, the only partition of {1, 2, 3} into 1 set can be 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: 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:

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


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

\( L(n,k) = {n-1 \choose k-1} \frac{n!}{k!} = {n \choose k} \frac{(n-1)!}{(k-1)!} \)
\( L(n,k) = \frac{n!(n-1)!}{k!(k-1)!}\cdot\frac{1}{(n-k)!} = \left (\frac{n!}{k!} \right )^2\frac{k}{n(n-k)!} \)
\( L(n,k+1) = \frac{n-k}{k(k+1)} L(n,k). \)
\( L(n,k) = \sum_{j} \left[{n\atop j}\right] \left\{{j\atop k}\right\} \), with \( \left[{n\atop j}\right] \) the Stirling numbers of the first kind, \( \left\{{j\atop k}\right\} \) 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

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

^ Introduction to Combinatorial Analysis Princeton University Press (1958, reissue 1980) ISBN 978-0691023656 (reprinted again in 2002, by Courier Dover Publications).

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