These conventions are used in combinatorics,[4] although Knuth's underline and overline notations
gives the number of n-permutations (sequences of distinct elements) from an x-element set, or equivalently the number of injective functions from a set of size
The coefficients that appear in the expansions are Stirling numbers of the first kind (see below).
is equal to the number of n-permutations from a set of x items, that is, the number of ways of choosing an ordered list of length n consisting of distinct elements drawn from a collection of size
is the number of different podiums—assignments of gold, silver, and bronze medals—possible in an eight-person race.
Equivalently, this is the number of ways to partition a set of size
The falling and rising factorials can be used to express a binomial coefficient:
Thus many identities on binomial coefficients carry over to the falling and rising factorials.
The rising and falling factorials are well defined in any unital ring, and therefore
Falling factorials appear in multiple differentiation of simple power functions:
Note, however, that the hypergeometric function literature typically uses the notation
Falling and rising factorials are closely related to Stirling numbers.
Indeed, expanding the product reveals Stirling numbers of the first kind
And the inverse relations uses Stirling numbers of the second kind
The falling and rising factorials are related to one another through the Lah numbers
are called connection coefficients, and have a combinatorial interpretation as the number of ways to identify (or "glue together") k elements each from a set of size m and a set of size n. There is also a connection formula for the ratio of two rising factorials given by
Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities:[11](p 52)
Finally, duplication and multiplication formulas for the falling and rising factorials provide the next relations:
The falling factorial occurs in a formula which represents polynomials using the forward difference operator
which in form is an exact analogue to Taylor's theorem: Compare the series expansion from umbral calculus with the corresponding series from differential calculus In this formula and in many other places, the falling factorial
in the calculus of finite differences plays the role of
A corresponding relation holds for the rising factorial and the backward difference operator.
The study of analogies of this type is known as umbral calculus.
A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences.
Falling and rising factorials are Sheffer sequences of binomial type, as shown by the relations:
Similarly, the generating function of Pochhammer polynomials then amounts to the umbral exponential,
[2] Graham, Knuth, and Patashnik[11](pp 47, 48) propose to pronounce these expressions as "
is typically used for the ordinary falling factorial, to avoid confusion.
and symbolic parameters x, t, related generalized factorial products of the form
may be studied from the point of view of the classes of generalized Stirling numbers of the first kind defined by the following coefficients of the powers of x in the expansions of (x)n,f,t and then by the next corresponding triangular recurrence relation: