In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
, Stirling numbers of the first kind arise in the study of permutations. In particular, the
Stirling number
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book ''Methodus differentialis'' (1730). They were rediscov ...
s of the first kind count
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s according to their number of
cycles (counting fixed points as cycles of length one).
The Stirling numbers of the first and
second kind can be understood as inverses of one another when viewed as
triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on
Stirling numbers in general.
Definitions
The original definition of Stirling numbers of the first kind was algebraic: they are the coefficients
in the expansion of the
falling factorial
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
:\begin
(x)_n = x^\underline &= \overbrace^ \\
&= \prod_^n(x-k+1) = \prod_^(x-k) \,.
\e ...
:
into powers of the variable
:
:
For example,
, leading to the values
,
, and
.
Subsequently, it was discovered that the absolute values
of these numbers are equal to the number of
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s of certain kinds. These absolute values, which are known as unsigned Stirling numbers of the first kind, are often denoted
or