Falling factorial

The terms of the falling factorial, in symbols

And the rising factorial, in symbols

Occur in combinatorics on in connection with a general definition of the binomial coefficient.

Definition and meaning

The falling factorial is defined as

It denotes the number of ordered samples from the periphery without repetition from one -element subset, or equivalently: the variation of elements to the class without repetition. For example, there are ways to distribute distinguishable balls to urns that no urn contains more than one ball.

The rising factorial is defined analogously:

The connection to the binomial coefficients create the following relationships:

In order to work with the factorial, the following relations are very helpful:

With the Faculty

It also apply the recursive relations:

Or more generally:

Notation

Usually, the symbol is used for the falling factorial. Since, however, can lead to confusion with a common notation for the Pochhammer symbol is sometimes also used the symbol. The same applies to the notation of the rising factorial.

Explicit formula

An explicit formula for the falling factorial is obviously given by: