[seqfan] Asymptotic pattern in columns of "A005646 triangle"

[Robert Munafo]

Franklin,

Your questions about the bottom of each column are interesting. I take it
that you are interested in seeing if there is asymptotic behaviour to the
bottom-most J terms of column J, or something like that. The result would be
similar to the appearance of the first J Catalan numbers (A000108) in row J
of the height-limited binary trees (A052154 and A137560, which are also
"Mandelbrot-related polynomials", see
http://mrob.com/pub/math/seq-a052154.html#trees )

Those columns definitely seem a bit hard to study computationally, as you
pointed out, because the relation N<=2^R allows N to grow so quickly. We
need to answer the questions:

   What are the most number of objects that can be classified with the union
of R binary partitions? (Simple, 2^R and in exactly one way)

   Considering the near-misses 2^R-1, 2^R-2, etc, and how many ways can sets
of each of those sizes be classified by R partitions?

(And thanks for the web-page correction, always appreciated!)

On Tue, Dec 29, 2009 at 06:21, <franktaw at netscape.net> wrote:

> Looking at your web site, it appears that the size 6 line graph is
> mislabeled.  It should be f-d-b-a-c-e, not f-e-d-c-b-a.
>
> It feels like I can almost prove this, but the final result eludes me.  It
> will suffice to prove that there must always (for R = N-1 > 0)  be one
> classification that distinguishes a single point.  The result then follows
> by induction - remove the point, find the graph for the remainder by
> induction, and there will be one point that is not distinguished from the
> removed point; join the edge to that point.
>
> -----
> I'm also interested in a question that is much less approachable with the
> development path you are following.  That is what happens at the "bottom" of
> each column -- just before it becomes zeros.  Does this have a limit?  If
> so, it would be 1,1,3,...., reading up the columns.  You would have to go to
> at least N=14 to get a clue to whether anything of this sort is happening
> (presumably the cases N=15 and N=16 can readily be shown to be 1).
>
> One can also ask about the column sums: 1,1,2,10,???
>
> Franklin T. Adams-Watters
>
> -----Original Message from: Robert Munafo
> Andrew Weimholt and I have been working on A005646. Franklin T.
> Adams-Watters suggested the importance of the triangle:
>
>   N=1   1
>   N=2   0  1
>   N=3   0  0  1
>   N=4   0  0  1  2
>   N=5   0  0  0  3   3
>   N=6   0  0  0  3  17    6
>   N=7   0  0  0  1  36   74   11
>   N=8   0  0  0  1  60  573  358   23
>   N=9   0  0  0  0  56 2802 7311 1631  47
> N=10  0  0  0  0  50 10087 107938 83170 7563 106
>
>  I've copied in row 10 from a later post - FTAW.
>>
>
> The main diagonal suggests a relation to sequence A000055, unlabaled
> unrooted trees.
>
> [...]
>


-- 
 Robert Munafo  --  mrob.com