Two ordering problems

[Max Alekseyev]

On Thu, Aug 28, 2008 at 4:05 PM,  <franktaw at netscape.net> wrote:
> I'm pretty sure that  http://www.research.att.com/~njas/sequences/A009997 is
> the desired sequence.  This can be interpreted as linear extensions of the
> lattice 2^n induced by assigning a weight to each primitive element, and taking the sum
> for the other nodes.  Taking the log of each divisor, we just need to match the
> relative ordering of the sums.
>
> I don't really see how to prove the final step, but the primes are pretty
> dense.  I would be astonished if any such arrangement cannot be obtained.

Unfortunately, there exist such arrangements (for n>=5).
Following the terminology of the paper
http://arxiv.org/abs/math.CO/9809134 (cited in A009997), an order on
the power set ("boolean term order") that is induced by some weights
of the singletons is called coherent. An example of noncoherent order
is given in Example 2.5 in the paper.

Sequence A009997 indeed gives the number of coherent boolean term orders but
for n>=5, its terms are smaller than the total number of boolean term
orders given by A005806.

Regards,
Max