Lah number
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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:
- <math> L(n,k) = {n-1 \choose k-1} \frac{n!}{k!}.<math>
Signed Lah numbers
- <math> L'(n,k) = (-1)^n {n-1 \choose k-1} \frac{n!}{k!}.<math>
Paraphrasing Karamata-Knuth notation for Stirling numbers it was proposed to use the following alternative notation for Lah numbers:
- <math>L(n,k)=\left\lfloor\begin{matrix} n \\ k \end{matrix}\right\rfloor.<math>