[seqfan] Re: A005646(11)=5236990, and triangle now linked to A039754

[Andrew Weimholt]

On Wed, Dec 30, 2009 at 9:24 PM, Robert Munafo <mrob27 at gmail.com> wrote:
> The A005646 "triangle" (which I am calling with my reserved A-number
> A171871) can be seen at mrob.com/pub/math/seq-a005646.html#triangle
>
> Thanks to analytical work by Andrew Weimholt we determined that the term
> just above the 1's at the bottom of each column is A034198, which led me to
> the paper at http://www.uni-graz.at/~fripert/blockcodes.html<http://www.uni-graz.at/%7Efripert/blockcodes.html>

Actually, it's the fourth to last non-zero term of the column.

The last 4 non-zero terms in column k are given by

A171871(2^k-3 , k) = A034198(k) for k>2
A171871(2^k-2 , k) = k for k>2
A171871(2^k-1 , k) = 1 for k>1
A171871(2^k , k ) = 1

> In Table 1 on page 3 we find:
>
> 1 1 1 1 1
> 1 1 1 1 1
> 1 2 3 4 5
> .. 1 3 6 10
> .. 1 6 19 47
> .. .. 3 27 131
> .. .. 3 50 472
> .. .. 1 56 1326
> .. .. 1 74 3779
> (etc.)
>
> which has many of the same numbers (reading down from the top) as the
> A171871 triangle reading up from the bottom. In the Nth column the first
> 2^(N-1) terms agree.

This is because A171871(n,k) can also be defined as the number of ways to
select n vertices on a k-dimensional cube (up tp rotation and
reflection) with the
restriction that at least one edge in each of the k directions must contain
two selected vertices. The vertices are the objects of the taxonomies, and
each dimension forms a binary partition, and the restriction
on the edges comes from the requirement that each partition be essential.
For the last 2^(k-1) terms in column k, the restriction has no effect
as more than
half of the vertices are selected making it impossible to violate the
restriction.
This is why the last 2^(k-1) terms agree with the table in the
block-codes paper.

The table in the can be transposed to generate a triangle and should be added
to the OEIS (when submissions are again allowed).

>
> Also, A005646(11)=5236990, and the whole N=11 row has been enumerated. The
> next row would likely take a few months with the current approach, so I'm
> not going to bother.
>
> But between Andrew and I, we have added 4 terms to a "nice,more" sequence
> that hasn't changed since 1979, and that's pretty good (-:
>

Yes it is. And it was a lot of fun too :-)

Andrew