User blog comment:XXxKennyDashxXx/Extended Separator Matrix Notation/@comment-25601061-20161007230342

Lemme guess.

If separator @ has level a, then a@b has growth rate w^a.

((0)) has level e_0

((0))(0) has level e_0+1

((0))(0)((0)) has level e_0*2

((0))(1) has level e_0*w

((0))(1)(2) has level e_0*w^w

((0))(1)((0)) has level e_0^2

((0))(2) has level e_0^w

((0))(2)((0)) has level e_0^e_0

((0))(3) has level e_0^(e_0*w)

((0))(4) has level e_0^e_0^2

...

((0))((0)) has level e_1

((0))((0))((0)) has level e_2

((0)1) has level e_w

((0)2) has level e_w^w

((0)(0)) has level e_e_0

Now, here's a problem with ((1)).

Is it either ((0)((0)((0)(...)))) or ((0)(0)(0)...(0)(0)(0))?

The former has growth rate gamma_0 while the latter has growth rate z_0.