# [seqfan] Re: Intersections of x^x^...^x

franktaw at netscape.net franktaw at netscape.net
Thu Nov 3 09:39:32 CET 2011

```It's perhaps a subtle point, but he asked for intersections of the
functions, not of the expressions. Expressions that produce the same
function are not considered different.

On a related note, I can't find the number of distinct function with n
x's in the database. I can find a number of cases where a particular
value of x is counted, but not the general case. Is it there and I'm
just not looking for it right? If it isn't there, it should be added.

-----Original Message-----
From: Andrew Weimholt <andrew.weimholt at gmail.com>

On Wed, Nov 2, 2011 at 7:10 PM, Vladimir Reshetnikov
<v.reshetnikov at gmail.com> wrote:
> Consider a set of functions obtained by all possible
parenthesizations of
> x^x^...^x (with n x's). How many pairwise intersections do they have
for
> 0<x<1? And for x>1?
>

for n>3, it is infinite...

(x^x)^(x^x) = (x^(x^x))^x

ln ((x^x)^(x^x)) = ln ((x^(x^x))^x)

(x^x) * ln (x^x) = x * ln (x^(x^x))

x*(x^x) * ln (x) = x*(x^x) * ln (x)

QED

Andrew

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```