Now we need to define arrays. Each array consists of a finite sequence of zero or more entries. Each entry consists of either a positive integer or another array (and these arrays can only nest finitely). An example of a valid array is
First, we define the following notation:
Hyperfactorial array notation defines a function n!A, where A is an array. An example of a well-formed expression in hyperfactorial array notation is 5![6, [7, 8], 9].
Define the active entry as the first entry in the array that is not 1. This is analogous to BEAF's pilot.
Define the receiving entry as the entry before the active entry, if the active entry is not the first entry.
The first entry is the main entry.
The symbol ◆ can be anything.
The symbol ◇ means only 1's and separators
- Any ones may be cropped off the end of an array:
- [@, 1] = [@]
- Any empty array can simply be replaced with n:
-  = n
- If the first entry is a number k>1:
- f(a) = a![k-1,@]
- n![k,@] = fn(n)
- n![1,k] = n![n,k-1]
- n![◇1,k@] = n![◇[◇1,1@],k-1@]
- Sometimes the active entry is an array. In that case:
- Use its main entry
- If the main entry is 1 reduce the array separately until either it produces a number or the main entry is no longer 1.
Where @ indicates the rest of the array.
(n) means next n-space. A comma is the same as (0). ▲ is any w/ chain. ▼ is an array with something before the first (k) divider. ▽ is an array without anything before the first (k) divider. ▬ is a string of ▽w(x)/'s for any x ◆ is any array or list of entries and separators. ▮ is a chain of ['s.
- ◆[▬▽w(k)/[q◆]▲]◆ = ◆[▬[1(k)1(k)1(k)...(k)1(k)2]w(k)/[1◆]▲]◆
- The ▽ in ◆[▬▽w(k)/[q◆]▲]◆ becomes [1(k)1(k)...1(k)2] with q copies of 1, q is replaced with 1, and the rest of the array does not change.
- Evaluate ▼ like this:
- When it is necessary to replace the receiving array with ▼ but the active entry changed to 1, or to make a w/ chain of ▼'s with the active entry decreased by one, use ▼w(k)/[q◆]▲ with the changes to ▼.
- Remove trailing 1's.
- If no k is specified for w(k)/, use 0, or w,/
HAN is quite new compared to other array notations, and its growth rate has not yet been agreed on. Hollom believes that it reaches all the way to the Takeuti-Feferman-Buchholz ordinal. n!n is smaller than the n+1th Ackermann number, and larger than the n-1th Ackermann number.
- Hollom, Lawrence. Hyperfactorial array notation. Retrieved 2015-02-27.