Lah number

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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 subsets and in such a way that all elements of the same subset are linearly ordered. 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)}
{(3, 2, 1)}

L(3, 2) corresponds to the 6 partitions with ordered parts:

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

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

[edit] See also

Stirling numbers

[edit] References

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