In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Lah numbers, discovered by Ivo Lah

Ivo Lah

Ivo Lah was a Slovenian mathematician and actuary, best known for his discovery of the Lah numbers in 1955.Lah attended elementary school in St. Vid nad Cerknico and in Ljubljana...

 in 1955, are coefficient

Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

s expressing rising factorials in terms of falling factorials.

Unsigned Lah numbers have an interesting meaning in combinatorics

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

: they count the number of ways a set of n elements can be partition

Partition of a set

In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...

ed into k nonempty linearly ordered subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

s. Lah numbers are related to Stirling number

Stirling number

In mathematics, Stirling numbers arise in a variety of combinatorics problems. They are named after James Stirling, who introduced them in the 18th century. Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second...

s.

Unsigned Lah numbers:


Signed Lah numbers:


L(n, 1) is always n!; using the interpretation above, the only partition of {1, 2, 3} into 1 set can be ordered in 6 ways:

{(1, 2, 3)}, {(1, 3, 2)}, {(2, 1, 3)}, {(2, 3, 1)}, {(3, 1, 2)} or {(3, 2, 1)}

L(3, 2) corresponds to the 6 partitions with two ordered parts:

{(1), (2, 3)}, {(1), (3, 2)}, {(2), (1, 3)}, {(2), (3, 1)}, {(3), (1, 2)} or {(3), (2, 1)}

L(n, n) is always 1: partitioning {1, 2, 3} into 3 non-empty subsets results in subsets of length 1.

{(1), (2), (3)}

Paraphrasing Karamata-Knuth notation for Stirling numbers, it was
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

with the Stirling numbers of the first kind, the Stirling numbers of the second kind and with the conventions and if .

Table of values

Below is a table of values for the Lah numbers:

	1	2	3	4	5	6	7	8	9	10	11	12
1	1
2	2	1
3	6	6	1
4	24	36	12	1
5	120	240	120	20	1
6	720	1800	1200	300	30	1
7	5040	15120	12600	4200	630	42	1
8	40320	141120	141120	58800	11760	1176	56	1
9	362880	1451520	1693440	846720	211680	28224	2016	72	1
10	3628800	16329600	21772800	12700800	3810240	635040	60480	3240	90	1
11	39916800	199584000	299376000	199584000	69854400	13970880	1663200	11880	4950	110	1
12	479001600	2634508800	4390848000	3293136000	1317254400	307359360	43908480	3920400	217800	7260	132	1