Lah number

In mathematics, the Lah numbers are the coefficients for the mutual representation of rising and falling factorials .

They were first described by Ivo Lah in 1955 . The following applies:

${\displaystyle x^{\overline {n}}=\sum _{k=0}^{n}L_{n,k}x^{\underline {k}}}$

The unsigned Lah numbers are defined as follows:

${\displaystyle L_{n,k}={n-1 \choose k-1}{\frac {n!}{k!}}}$

The signed Lah numbers are defined by

${\displaystyle L_{n,k}=(-1)^{n}{n-1 \choose k-1}{\frac {n!}{k!}}}$

The unsigned Lah numbers are used for the inversion formula of the rising and falling factorials.

In which they have combinatorics an interesting property: they describe the number of linearly ordered - partitions a -element amount . ${\displaystyle k}$ ${\displaystyle n}$

In addition:

${\displaystyle L(n,k)=B_{n,k}(1!,2!,3!,4!,\ldots )}$

where stands for the Bell polynomials . ${\displaystyle B_{n,k}}$

values

${\displaystyle _{n}\!\!\diagdown \!\!^{k}}$	1	2	3	4th	5	6th	7th	8th	9
1	1
2	2	1
3	6th	6th	1
4th	24	36	12	1
5	120	240	120	20th	1
6th	720	1800	1200	300	30th	1
7th	5040	15120	12600	4200	630	42	1
8th	40320	141120	141120	58800	11760	1176	56	1
9	362880	1451520	1693440	846720	211680	28224	2016	72	1

(Follow A008297 in OEIS )

Individual evidence

1. ↑ ^{a b} Eric W. Weisstein: Lah Numbers on MathWorld