Lah number

The Lah numbers in mathematics, the coefficients for the mutual representation of rising and falling factorials.

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

The unsigned Lah numbers are defined as follows:

The unsigned Lah numbers are defined by

For the inversion formula of the rising and falling factorials to use the unsigned Lah numbers.

These have in combinatorics an interesting property: they describe the number of linearly ordered partitions of a - element set.

In addition:

Where stands for the Bell polynomials.

Values

( Sequence A008297 in OEIS )