[seqfan] Re: intersection patterns of n sets

[Robert Munafo]

I think that Arndt's (*) is worth mentioning from whichever core seq. it
turns out to be, and I also think it deserves a cross-reference from
A005646.

I also think (*) may be related to A005646, at least in a semantic way
(they evoke similar ideas). I don't think there's much more than that,
though it seems to me that Arndt's (*) is kind of like the answer to the
question "how many cases in the A005646 problem end up involving n
partitions" except that in the case of A995646 each partition must be
accomplishing something, i.e. it divides a set into two nonempty disjoint
subsets, whereas with (*) the partition implicit in "membership of a set"
might include everything, i.e. its complement is the null set.

Anyway, the table at http://mrob.com/pub/seq/a005646.html#triangle shows
distinct partitioning schemes organized by number of partitions (R) and
number of categories (N) counting the nonempty disjoint sets into which the
partitions classify the world. In the a005646 problem one of the "A
disjoint from B" cases is equivalent to one of the "one set is a subset of
the other" cases. Also, all "set A = set B" cases are excluded because then
you'd have two partitions that do the same thing, so one is
redundant; and the "set A includes everything" cases are excluded because
set A would represent a partition that accomplishes nothing; keeping in
mind that the labels A and B are not significant so "A subsetof B
subsetof Universe" counts as the same case as "B subsetof A subsetof
Universe". Also anything involving a null set, such as "set A = set B =
nullset" but that's just a special case of "A=B" already mentioned; and the
complement of (union of all the sets) must be non-null but that's the same
as excluding the "A=Universe" cases.

(Sorry for wordiness and lack of canonical terminology.)


> From: Joerg Arndt <arndt at jjj.de <javascript:;>>
>
> Removing all "geometric" restrictions from  https://oeis.org/A250001 and
> changing "circles" to "sets" we get a comment that could possibly put into
> some(*) sequence:
> "a(n+1) is the number of intersection patterns of n (unlabeled) sets of
> unlabeled elements."
> [...]

"Take n nodes for the sets and put an edge between to sets whenever they
> have a nonempty intersection; take one extra node and put an edge to a set
> whenever it has an element that is unique to it."
>
> Yes, sequence (*) is  https://oeis.org/A000088
> Is this a comment worthy of being put into A000088 ?
> If so, should a similar comment appear in  https://oeis.org/A001349 ?
>

From: Neil Sloane <njasloane at gmail.com <javascript:;>>
>

> With 2 sets, they are disjoint, or one contains the other, or they have a
> nontrivial intersection, which is three ways, and 3 is not a term of
> A000088.
>
> So I don't understand your remark.
>
> Perhaps you were thinking the sets could also be equal? But that does not
> seem fair.  If you start with three sets, then they are all different




From: Joerg Arndt <arndt at jjj.de <javascript:;>>
>
> Perhaps you were thinking the sets could also be equal?
>
> Yes, of course!
>
> > But that does not seem fair.
>
> I do not understand why that should be bad: A <==> B  is something I'd
> very much like to allow.
>
> But I realize that I did not think it through: a set not connected to any
> other would be forced to be connected to that "extra set".
> Anyway, I speculate this is one of the old "core" seqs.
>


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