Inverse power tower functions
I consider the functional inverse of power towers made with x and 2.

These functions are imho fundamental ; they are the seed for brute asymptotes to most real-entire functions.

Height 2

Inv x^2
Inv 2^x
Inv x^x

With their resp solutions : sqrt , binairy log , ssqrt.
Notice the ssqrt can be expressed by ln and lambertW.

Most strictly rising functions that grow slower then x already grow like logs , powers and ssqrt type functions ( by finite composition , addition and product ).

I came to consider the enumeration of these functions.

For height 2 , as shown above , we have 3 functions.

The pattern seems simple , but might not be.

How many functions do we have Up till height 3 ?

You probably guessed 6 if you are fast.

But it is 5.


Y = x^ ( x^2)

Ln(y) = ln x x^2.

This equation can be solved by LambertW ( just like x^x = y could ).

So we get 3 for height 2 and max 2 extra for height 3 :

Inv x^x^x
Inv x^2^x

Giving a Total of max 5.

The fact that x^x^2 reduces is domewhat surprising.

So care is needed.

So for instance how many fundamental function do we have Up to height 17 ?

A related question is how Some of them make good asymptotics of others and Some do not.

For instance ssqrt has asymptotics in terms of the others of height 2 ( logs and powers ).

Conjectures are easy to make for the amount of functions Up to Some height
For instance

5 + 2^(h-3) for hights h >=3.

But hard to prove.




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