[seqfan] Re: Help wanted on A005646 connection to A000055

[Robert Munafo]

Andrew Weimholt completed a longer, more elaborate and thorough proof and
sent it to me several hours earlier. I believe it covers the full problem of
showing that the two integer sequences are equal.

Franklin, I believe you have part of it but not the whole thing... I'm
pretty fuzzy on formal logic, so we should ask Andrew.

As Andrew described it to me, one must show that each tree corresponds to an
(N,R=N-1) classification, that each such classification corresponds to a
tree, and that the equivalence of any two given classifications implies that
their corresponding trees are also equivalent. Each of these parts had
sub-parts, he used 7 intermediate theorems.

In other news, I have computed the 4th column of the triangle fully, posted
separately.

On Tue, Dec 29, 2009 at 18:00, <franktaw at netscape.net> wrote:

> I can now prove that the diagonal is indeed A000055.
>
> Clearly each tree gives rise to a classification with the desired property,
> with each edge being mapped to a binary partition distinguishing between
> points in the two components created when the edge is removed.  We need to
> show that any such classification arises from such a tree.
>
> Given a classification of N points with R = N-1 binary partitions, choose
> any of the binary partitions.
>

>
[...]
>

> P.S. I sent the sub-diagonal 1,3,17,74,358,1631,7563 to superseeker; it
> found nothing.
>

>
Robert Munafo wrote:
>

>
> The main diagonal suggests a relation to sequence A000055, unlabaled
> unrooted trees.
>
> > I am starting to convince myself such a connection is real; detals and
> full examples through N=6 are athttp://mrob.com/pub/math/seq-a005646.html#trees
>

-- 
 Robert Munafo  --  mrob.com