Factorials
Factorials are a fundamental component to combinatorics.
Description
A factorial is a product: x! = 1 * 2 * ... * x. This represents the number of possible samples without replacement.
Rising Factorials
A rising factorial is a related concept. It takes two arguments, x and n, and it expands to x * (x+1) * ... * (x + n - 1).
A rising factorial is sometimes notated as (x)n, but to avoid ambiguity the following is recommended: 
Rising factorials are polynomial series characterized by Stirling numbers of the first kind. Consider the following example:
Stirling numbers of the first kind provide a generic expansion of any rising factorial:
Stirling numbers of the first kind are ordinarily signed, so the absolute value is taken in the above formulation.
Falling Factorials
A falling factorial is another related concept, this time expanding to x * (x-1) * ... * (x - n + 1).
A falling factorial is sometimes notated as (x)n, but to avoid ambiguity the following is recommended: 
Falling factorials are also polynomial series characterized by Stirling numbers of the first kind:
In particular, note that these coefficients are identical except for the signs. The generic expansion of any falling factorial is:
where the absolute value is not taken.
Stirling Numbers
Stirling numbers of the first kind give the coefficients for polynomial series expansions of rising and falling factorials. As shown above, they are notated s(n,k).
Unsigned Stirling numbers of the first kind can be written in a triangular matrix much like Pascal's triangle. They represent the number of permutations of n elements into k disjoint cycles. The signs can easily be derived as (-1)n-k.
As the phrasing alludes to, there are also Stirling numbers of the second kind. These are notated S(n,k) and refer to the number of ways to partition a set of n elements into k non-empty subsets. More directly, these represent an inverse relation: expressing a series of falling factorials as a polynomial.
To express in terms of rising factorials, signs have to be reintroduced using the same mechanism as above.
