# (x^x)^(x^x), x^(x^(x^x)), etc...

Meeussen Wouter (bkarnd) wouter.meeussen at vandemoortele.com
Tue Apr 29 11:11:26 CEST 2003

```specialising x to Sqrt[2], only these remain:
1,2,4,8,17,38,88,206,498 ...

Wouter.

-----Original Message-----
From: Edwin Clark [mailto:eclark at math.usf.edu]
Sent: maandag 28 april 2003 17:30
To: seqfan
Cc: math-fun at mailman.xmission.com
Subject: (x^x)^(x^x), x^(x^(x^x)), etc...

Consider the following five functions defined on positive reals:

f1(x) = ((x^x)^x)^x;

f2(x) = (x^(x^x))^x;

f3(x) = x^((x^x)^x);

f4(x) = x^(x^(x^x ));

f5(x) = (x^x)^(x^x);

Note that f2 = f5. There are really only 4 different
functions of this type. We know the Catalan number C(n-1)
gives the number of ways to parenthesize a product of
n variables. If we use x^x for the binary operation and identify
products that give equal functions when x is positive,
how many products do we get?

If the number of x's is n let e(n) be the number of distinct
functions so obtained. I used Maple to simplify the set
of C(n-1) expressions and got the following cardinalities
for the simplified sets for n from 1 to 11:

(*)     1,1,2,4,9,20,48,115,286,719,1842

Except for some of the smaller values, I cannot be sure that
Maple has not failed to identify some equal functions. So
this may not be the sequence e(n).

The sequence (*) coincides with A000081: rooted trees with n
nodes --- at least as far as I have computed. Is this known?
If not can anyone prove that the sequences coincide?

Maple undoubtedly uses standard laws of exponents to simplify
these expressions. This in itself raises a number of questions.

Edwin Clark

===============================
This email is confidential and intended solely for the use of the individual to whom it is addressed.
If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited.
You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

```