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Template:Function Hyperfactorial array notation is a large number notation invented by Lawrence Hollom. It was first developed in April 2013.

## Basics Edit

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

[1,1,[1,2,,4,[],1,1],1,3,10,1,[4,[4,3,1],5,6],1,[1,2],1,1]

First, we define the following notation:

\(n!m = n\uparrow^{m}(n-1)\uparrow^{m}(n-2)\cdots 4\uparrow^{m} 3 \uparrow^{m} 2 \uparrow^{m} 1\)

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 \(5![6, [7, 8], 9]\).

## Linear arrays Edit

Define the active entry as the first entry in the array that is not 1. This is analogous to BEAF's pilot.

• 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 \(k>1\):
\(f(a) = a![k-1,@]\)
\(n![k,@] = f^n(n)\)
• Otherwise:
\(n![1,1,\cdots,1,1,[[...[[k @]]...]],@]\)
\( = n![1,1,\cdots,1,[1,1,\cdots,1,1,[[...[[1 @]]...]],@],[[...[[k-1 @]]...]],@]\)

Here \(@\) indicates the rest of the array.

## Analysis Edit

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.

## Sources Edit

1. Template:Citation/CS1

## Edit

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