Lah number

In mathematics, Lah numbers, discovered by Ivo Lah in 1955 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 that 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

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

Paraphrasing Karamata-Knuth notation for Stirling numbers it was proposed to use the following alternative notation for Lah numbers:

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

See also

Stirling numbers