User blog comment:Eners49/New Notation Idea? (Attempt 2)/@comment-35392788-20180709175008

First, let me extend your notation, by going off what you've already done.

{n} = ((...(n!)!...)!)! where there are n factorial signs

{#,1} = {#}, where # is the rest of the array.

{a,b+1,#} = {{...{a}...},b,#} where there are a sets of brackets

{a,1,...,1,c+1,#} = {a,1,...,a,c,#}

~ means "corresponds to FGH ordinal"

{a} ~ 3

{a,2} ~ 4

{a,3} ~ 5

{a,a} ~ w

{a,1,2} ~ w

{a,2,2} ~ w+1

{a,1,3} ~ w*2

{a,2,3} ~ w2+1

{a,1,a} = {a,1,1,2} ~ w^2

{a,2,1,2} ~ w^2+1

{a,1,2,2} ~ w^2+w

{a,1,3,2} ~ w^2+w*2

{a,1,1,3} ~ w^2*2

{a,1,1,4} ~ w^2*3

{a,1,1,1,2} ~ w^3

The limit of this notation is w^w, just like linear BEAF.