[seqfan] Re: connectedness systems: the conjecture about A072447

[Christian Sievers]

Dear Neil , dear list!

On Sun, Oct 22, 2023 at 03:57:05PM -0400, Neil Sloane wrote:
 
> May I suggest that you take over the editing of this sequence, and make any
> changes that you think are appropriate?

I have done that now (not proposed for review yet).

I saw two options for the definition: the old one, or replacing it with
something that talks about connectedness systems (and doing the same
with related sequences).

Since apparently changing the definition is acceptable, and the old one
is not really correct (because of the issue with the empty set as
element of the families), I chose the second option.

I also considered using "connected connectedness systems" instead of
just adding the condition that it must contain the full set. I decided
against using "connected" because, unlike "connectedness system", it
doesn't appear very often and isn't much of an abbreviation. 
(Also, the definition given in the comment seems a bit odd. Why does it
allow the empty familie? With the suggested definition that allows no
singletons, that would add the empty familie for all n.)

I used all singletons instead of none so that a(1)=1 without extra
complications. Note also that this definition would give a(0)=0, as
we'd want for the exponential transform to A072446.

Since the current definition of connectedness system is a bit hidden, I
repeated it at the beginning of the comments. Not sure if that's the
best way to handle this.
(While I understand the culture of keeping track who contributed what,
editing Wikipedia-style and freely moving stuff around and rewriting it
[which I haven't done for years] would be so much easier.)

In the remark to the conjecture, I replaced
"The new definition makes it clear that the conjecture does not hold.
See" with
"It's easy to see that for n>1, a(n) is also the number of
families of subsets of {1, ..., n} that contain both the universe and
the empty set, are closed under union of nondisjoint sets, and contain
no singletons; whereas by duality, the sequence suggested in the
conjecture is also the number of those families that are also closed
under arbitrary union. For details see",
which gives the idea and is hopefully not too long.
It could be shorter if we applied the duality to the alternative
description of a(n) (because we wouldn't have to talk much about the
conjectured version) but that would distance it from my linked note.


I also changed A072446 to have the corrected value of the sixth term and
have a definition using "connectedness systems".
It sounds a bit odd because the definition that the comments already
have is conditionally phrased: "If we define a connectedness system to
be ...", but I left it like that.

On A326866 I just changed the sixth term. It's definition already used
connectedness systems.

Even though these changes are not yet proposed for review, there are
already doubts and a wish for independent confirmation.
I agree with that, that's why I wrote my other mail in the first place.

> If you think the best way to proceed is to have two (or even three)
> sequences, then please go ahead and propose them.

I see no need for further sequences in this case.

> Send me an email when you have made the changes. (I am in the middle of
> several urgent projects, with deadlines coming up, so I have less time than
> usual for monitoring the submissions.)

Good luck for your projects!


All the best
Christian