Lah number


In mathematics, the Lah numbers, discovered by Ivo Lah in 1954, are coefficients expressing rising factorials in terms of falling factorials. They are also the coefficients of the th derivatives of.
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 linearly ordered subsets. Lah numbers are related to Stirling numbers.
Unsigned Lah numbers :
Signed Lah numbers :
L is always n!; in the interpretation above, the only partition of into 1 set can have its set ordered in 6 ways:
L corresponds to the 6 partitions with two ordered parts:
L is always 1 since, e.g., partitioning into 3 non-empty subsets results in subsets of length 1.
Adapting the Karamata–Knuth notation for Stirling numbers, it has been proposed to use the following alternative notation for Lah numbers:

Rising and falling factorials

Let represent the rising factorial and let represent the falling factorial.
Then and
For example,
Compare the third row of the table of values.

Identities and relations

Table of values

Below is a table of values for the Lah numbers:
123456789101112
11-----------
221----------
3661---------
42436121--------
5120240120201-------
672018001200300301------
7504015120126004200630421-----
84032014112014112058800117601176561----
936288014515201693440846720211680282242016721---
1036288001632960021772800127008003810240635040604803240901--
1139916800199584000299376000199584000698544001397088016632001188049501101-
12479001600263450880043908480003293136000131725440030735936043908480392040021780072601321