notation

Variations on factorial!

Posted on by John

If you’ve heard of factorial, have you heard of double factorial or subfactorial?

Double factorial is written n!!. The factorial of a positive integer n is the product of all positive integers less than or equal to n. The double factorial of n is the product of all integers less than or equal to n that have the same parity.  That is, for an odd number n,  the product defining n!! includes only odd integers and for an even integer n, the product defining n!! includes only even integers. For example, 7!! = 7 × 5 × 3 × 1 and 8!! = 8 × 6 × 4 × 2. By definition, 0!! and (−1)!! equal 1.

Double factorials often arise in integrals and power series and make it possible to state equations succinctly that would be verbose otherwise. For example,

\int_0^{\pi/2} \sin^{2n+1} \theta , d\theta = \frac{(2n)!! }{ (2n+1)!!}

It’s possible to define higher factorials or multifactorials. For instance n!!!, the triple factorial of n, is the product of positive integers less than or equal to n and congruent to n mod 3. So, for example, 8!!! = 8 × 5 × 2.

Factorials count the number of ways a set can be arranged. A set with n distinct elements can be arranged in n! ways. The number of arrangements that move every element from its original position is the subfactorial of n. Sometimes subfactorial is written with the exclamation point in front of its argument and sometimes it is written with an inverted exclamation point following its argument, i.e.

!n = nmbox{!`}

(By the way, the inverted exclamation mark, used to mark the beginning of an exclamatory sentence in Spanish, is Unicode character U+00A1. You can produce it in HTML with &iexl;. In TeX, you can produce it !` outside of math mode and mbox{!`} in math mode.)

Subfactorial can be computed from the factorial by

nmbox{!`} = \leftl\floor \frac{n!}{e} + \frac{1}{2} \right\rfloor

for positive n where ⌊x⌋ is the greatest integer less than x. The subfactorial of 0 is defined to be 1.

Update: See post on mutifactorials.

Related posts

Four uncommon but handy math notations

Posted on by John

Here are some of my favorite notations that are not commonly used.

The first is Richard Stanley’s notation for counting the number of ways to select k objects from a set of n objects with replacement. This is similar to the problem solved by binomial coefficients, but not the same since binomial coefficients count the number of possible selections without replacement. Stanley’s symbol is

\left( {n \choose k} \right)

I like this symbol for two reasons. First, it’s good to have a notation, any notation, for a concept that comes up fairly often. Second, it’s appropriate for this symbol to resemble the binary coefficient symbol. See selecting with replacement for more on Stanley’s symbol, how to think about it and how to compute it.

Next is Kenneth Iverson’s notation for indicator functions. Iverson’s idea was to put a Boolean condition in square brackets to indicate the function that is 1 when that condition is true and 0 otherwise. For example, [x > y] is the function f(x, y) such that f equals 1 when x is greater than y and equals 0 for all other arguments. This notation saves ink and makes it easier to concentrate on the substance of an expression. For more on Iverson’s notation, see Concrete Mathematics.

Another notation from Concrete Mathematics is the use of a perpendicular symbol to note that two integers are relatively prime. For example, m ⊥ n would indicate that m and n are relatively prime. The more common way to denote this would be to say gcd(m, n) = 1. The perpendicular symbol is nice because perpendicular lines have no component of direction in common, just as relative prime numbers have no prime factors in common.

Finally, multi-index notation is a handy way to make multivariable theorems easier to remember. For example, with this notation, Taylor series in several variables look similar to Taylor series in one variable.

Related link: Stanley’s twelvefold way