[seqfan] Re: Intersections of x^x^...^x
Alonso Del Arte
alonso.delarte at gmail.com
Thu Nov 3 18:23:40 CET 2011
I knew that sequence was already in the OEIS. Thank you for finding it,
Alois. I've made A81 the Sequence of the Day for January 5 and some of
Vladimir's sequences for specific x's for some other days.
P.S. Even if I wanted to write the Sequence of the Day for every single
day, I couldn't.
On Thu, Nov 3, 2011 at 6:53 AM, Alois Heinz <heinz at hs-heilbronn.de> wrote:
> See this comment from A000081:
> Take a string of n x's and insert n-1 ^'s and n-1 pairs of parentheses
> in all possible legal ways (cf. A003018). Sequence gives number of
> distinct functions. The single node tree is "x". Making a node f2 a
> child of f1 represents f1^f2. Since (f1^f2)^f3 is just f1^(f2*f3) we
> can think of it as f1 raised to both f2 and f3, that is, f1 with f2
> and f3 as children. E.g. for n=4 the distinct functions are
> ((x^x)^x)^x; (x^(x^x))^x; x^((x^x)^x); x^(x^(x^x )). - Edwin Clark
> (eclark(AT)math.usf.edu) and Russ Cox (rsc(AT)swtch.com) Apr 29, 2003;
> corrected by Keith Briggs (keith.briggs(AT)bt.com), Nov 14 2005
> See also comment in https://oeis.org/draft/A199085
> Vladimir Reshetnikov: A000081 gives a number of distinct function that
> can be obtained by parenthesizing x^x^...^x. So, it is an upper bound
> for this sequence.
> Am 03.11.2011 09:39, schrieb franktaw at netscape.net:
> 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.
>> Franklin T. Adams-Watters
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Alonso del Arte
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