[seqfan] Re: Interesting sequence needs one more term (lattices of subsets)

[Andrew Weimholt]

Hi Neil,

I've computed the number of realizable equivalence classes for n=5, and got
7443.
My program also confirmed 50 for n=4.

Instead of constructing the equivalence classes first, I construct the
realizations
then compute the equivalence class, test to see if it is canonical, and if
so,
I make sure I haven't already counted it, and if it's new, I increment my
count.

There are 2^n ways in which an element can belong to n labeled sets.
We can throw out the element that belongs to no sets, and the one that
belongs
to all n sets, as those two elements do not help us distinguish one
collection of
sets from another. This leaves us with 2^n-2 elements to play with, and
every subset of these 2^n-2 elements forms a realization of some equivalence
class. This gives me 2^(2^n-2) realizations to examine (2^30 for n=5).

Next I'll write a program which constructs the equivalence classes first,
without caring if they're realizable. I suspect that all equivalence classes
with non-contradictory inclusion relations will be realizable, and the
number should
come out the same.

Andrew



On Tue, Jan 21, 2014 at 7:32 PM, Neil Sloane <njasloane at gmail.com> wrote:

> Dear Seq Fans, http://oeis.org/A235604 looks very interesting,
> so I created an entry for it even though there
> are already many sequences that start the same way
> (1,4,50). If we had one more term, it might settle
> the question of whether it is really new. My guess is that it will be new.
> Neil
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> Neil J. A. Sloane, President, OEIS Foundation
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
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