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Topic: Stone Cech
Replies: 49   Last Post: Mar 28, 2013 12:15 PM

 Messages: [ Previous | Next ]
 Butch Malahide Posts: 894 Registered: 6/29/05
Re: Stone Cech
Posted: Mar 20, 2013 5:47 PM

On Mar 20, 8:53 am, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> On Tue, 19 Mar 2013 21:25:44 -0700 (PDT), Butch Malahide
> <fred.gal...@gmail.com> wrote:

> >On Mar 19, 8:49 pm, William Elliot <ma...@panix.com> wrote:
> >> On Tue, 19 Mar 2013, David Hartley wrote:
> >> > > > > Perhaps you could illustrate with the five different one to four point
> >> > > > > point compactifications of two open end line segements.

>
> >> > > > (There are seven.)
> [. . .]
> >> Ok, seven non-homeomophic finite Hausdorff compactications.
>
>
> Surely there's no simple formula for that?
>
> The "obvious" approach seems like starting with the number
> of partitions of 2n, which is already hopeless, and then

The number of partitions of 2n? As noted by Euler, p(2n) is just the
coefficient of z^{2n} in the Maclaurin expansion of the function F(z)
= 1/eta(z), where eta(z) is the infinite product
eta(z) = (1 - z)(1 - z^2)(1 - z^3)...(1 - z^n)....
From there you can work out the asymptotics etc. The recurrence
relation is
p(n) = p(n-1) + p(n-2) - p(n-5) - p(n-7) + p(n-12 + p(n-15) - p(n-22)
- ...

> somehow factoring out the ones that give homeomorphic
> compactifications... hopeless squared.

In graphical terms, it's about enumerating the nonhomeomorphic graphs
(undirected but with loops and multiple edges permitted) with n edges.
No doubt this is much harder than the (solved) enumeration problem for
nonisomorphic simple graphs with n vertices. It's just not immediately
obvious to me that the question is "hopeless squared". Of course it's
way too hard for me; that's why I posted the question instead of
trying to figure it out myself.

> Is there a better way, or are you just making trouble again or what?

I wasn't necessarily expecting a *complete* answer, such as an
explicit generating function. Maybe someone could give a partial
answer, such as an asymptotic formula, or nontrivial upper and lower
bounds, or a reference to a table of small values, or the ID number in
the Encyclopedia of Integer Sequences, or just the value for n = 3. (I
got 21 from a hurried hand count.)

Date Subject Author
3/14/13 William Elliot
3/14/13 fom
3/15/13 fom
3/16/13 William Elliot
3/15/13 David C. Ullrich
3/17/13 William Elliot
3/17/13 David C. Ullrich
3/17/13 fom
3/18/13 David C. Ullrich
3/18/13 fom
3/18/13 David Hartley
3/19/13 William Elliot
3/19/13 David Hartley
3/19/13 William Elliot
3/20/13 Butch Malahide
3/20/13 David C. Ullrich
3/20/13 Butch Malahide
3/20/13 Butch Malahide
3/21/13 quasi
3/21/13 quasi
3/21/13 quasi
3/21/13 quasi
3/21/13 Butch Malahide
3/21/13 quasi
3/22/13 Butch Malahide
3/22/13 Butch Malahide
3/22/13 Butch Malahide
3/22/13 quasi
3/22/13 David C. Ullrich
3/22/13 David C. Ullrich
3/22/13 Butch Malahide
3/23/13 Butch Malahide
3/23/13 David C. Ullrich
3/23/13 David C. Ullrich
3/23/13 Frederick Williams
3/23/13 David C. Ullrich
3/23/13 Frederick Williams
3/22/13 Butch Malahide
3/23/13 David C. Ullrich
3/22/13 Butch Malahide
3/23/13 quasi
3/23/13 Butch Malahide
3/23/13 Butch Malahide
3/24/13 quasi
3/24/13 Frederick Williams
3/24/13 quasi
3/25/13 Frederick Williams
3/28/13 Frederick Williams
3/25/13 quasi
3/19/13 David C. Ullrich