10,973 Pages

Left- or right-hand parenthesisation? Former one is I think stronger, and is easier to formally define. LittlePeng9 (talk) 18:52, November 17, 2013 (UTC)

## Growth Rate

The growth rate is $$\omega$$. —Preceding unsigned comment added by Googleaarex (talkcontribs)

Yes, indeed. You can add Template:function. LittlePeng9 (talk) 19:45, November 17, 2013 (UTC)

## Extension

Make an extension of Iterated Mixed Factorial.

Then make an extension of this, then extension of this, then extension of this, ...AarexTiao 22:47, November 17, 2013 (UTC)

Array Mixed Factorial: a*[b,c @] = :a*[c*...[c @]... @] (b c's) AarexTiao 01:26, November 21, 2013 (UTC)
[a,b#c] = [a,b:2c]. So we make a extension, not above. AarexTiao 16:45, November 23, 2013 (UTC)
...Then make an extension of this (7th), then extension of this (8th), then extension of this (9th), ... up to 77th extension. AarexTiao 00:54, November 24, 2013 (UTC)

## Formal ruleset and growth rate

For basic mixed factorials, we can just take:

1. $$n\text{*} = 1+2\times3\uparrow4\uparrow^25...n-2\uparrow^{n-4}n-1\uparrow^{n-3}n$$

Growth rate: $$\omega$$

First extension:

2. $$n\text{*}^k = n\underbrace{\text{**...**}}_k$$

3. $$n\text{*}^{[k,x\text{:}1]} = n\text{*}^{x\text{*}^{x\text{*}^{...}}}$$

Growth rate: $$\omega+2$$

Second extension:

4. $$n\text{*}^{[k,x\text{:}y+1]} = n\text{*}^{[n\text{*}^{[n\text{*}^{...},x\text{:}y]},x\text{:}y]}$$

Growth rate: $$\omega2$$

Third extension:

5. $$n\text{*}^{[k,x\#2]} = n\text{*}^{[k,x:n\text{*}^{[k,x]}]}$$

6. $$n\text{*}^{[k,x\#y+1]} = n\text{*}^{[k,x:n\text{*}^{[k,x\#y]}]}$$

Growth rate: $$\omega2+1$$

Fourth extension:

7. $$n\text{*}^{[k,x\#_y\text{|}2]} = n\text{*}^{[k,x\#_{y-1}\text{|}n\text{*}^{[k,x]}]}$$

8. $$n\text{*}^{[k,x\#_{z}\text{|}y+1]} = n\text{*}^{[k,x\#_{z-1}\text{|}n\text{*}^{[k,x\#_{z}\text{|}y]}]}$$

Growth rate: $$\omega3$$

Fifth extension:

Previous rules also apply if there are arrays.

$$\bullet$$ is any array

$$\diamond$$ is a string of ones.

9. $$n\text{*}^{[k,x\#_{\diamond,1,2,\bullet}\text{|}2]} = n\text{*}^{[k,x\#_{\diamond,n\text{*}^{[k,x\#2]},1,\bullet}\text{|}n]}$$

10. $$n\text{*}^{[k,x\#_{\diamond,1,2,\bullet}\text{|}y+1]} = n\text{*}^{[k,x\#_{\diamond,n\text{*}^{[k,x\#_{\diamond,1,2,\bullet}\text{|}y]},1,\bullet}\text{|}n]}$$

S1. Zeroes at the end of an array can be removed.

Growth rate 2-entry: $$\omega^2$$

Growth rate: $$\omega^\omega$$

Wythagoras (talk) 07:57, November 24, 2013 (UTC)

How about 6th extension ruleset and growth rate? AarexTiao 15:07, November 24, 2013 (UTC)

## Unsourced

I commented out some of the unsourced content. When it's in the source, feel free to uncomment it again. FB100Ztalkcontribs 20:25, November 24, 2013 (UTC)

No! Don't uncomment until you made the page. AarexTiao 01:46, November 25, 2013 (UTC)
Do we really need to keep these stuff on this article in hidden comments? The extension is not yet been added outside of Googology Wiki (even Aarex says so), and doesn't belong in articles. That's what user pages and blogs are for. Furthermore, the edits on the commented out stuff on this page makes searching for "legitimate" edits harder. So we should remove the commented out content and ask SpongeTechX to add a definition of the extension on the Googology World, so we could add it back. -- ☁ I want more clouds! ⛅ 02:41, November 28, 2013 (UTC)

I honestly think I might need to completely restart the extension and delete all numbers that have to do with the extension. Aarex kind of just did what he wanted with it anyway. :\
07:52, November 28, 2013 (UTC)

## New Page

Please add a page, called Extended Mixed Factorial, that all extensions is same at all Mixed factorial extensions, in Googology World. AarexTiao 21:19, November 24, 2013 (UTC)

Community content is available under CC-BY-SA unless otherwise noted.