W.E.A.K. (Whimpy Effete Array of K.Kebab)
The purpose of this notation is to create the smallest possible Array notation while still producing numbers that are not used in everyday terms (and to jokingly respond to this blog post :P)
Base case
(a) turns into a+1
(a, b) turns into ((a, b-1), b-1)
and (a, b, c, d, ... x, y,) turns into (a, b, c, d, ... (x, y-1), y-1)
To ensure the entries terminates to give a finite value we can define a closing case :
(a, b, c, d, ... x, 1) turns into (a, b, c, d, ...x)
And as follows
(#, 1, @) turns into (#)
where # and @ denotes valid arrays.
This is not very powerful, exactly what I had in mind. It is easy to show that (a, b) = a + 2^{(b-1)} and as such it is easy to prove that (a, b) > \(f_2(b)\) given that a is smaller than b.
Next, to diagonalize over the linear arrays, we define (a[0]) as (a, a, a, ... a, a) with a number of arrays. It is again easy to show that (a[0]) > \(f_3(a)\). To further extend it define (a[n+1]) to be (a[n], a[n], a[n], ... a[n]) with a a[n]'s. By looking at the definition I conjecture that (a[a]) > f_omega(a) (using the fundamental sequence [0, 1, 2, ...] for omega) , but a better lower bound would be f_(a-3)(a) for all a >2.
This notation is so small that it invokes no large limit ordinals and thus no (maybe not) confusion involving fundamental sequences can arise from this.
(P.S. this blog post is a joke)