[seqfan] Re: EIS Mistake on the Dedekind combinatory sequences.

[M. F. Hasler]

On Mon, May 31, 2021 at 2:24 PM Sean A. Irvine <sairvin at gmail.com> wrote:

> The sequence is defined to be the first differences of A132581 and the
> values are correct.
> Robert Israel's Maple program correctly computes the sequence.
>
I agree.


> The "n>=1" in the comment is purely a restriction about the region in which
> that comment applies.
>

Yes but there was some confusion.
I think the comment should indeed have "n" instead of "n-1",
on the other hand it is correct for all n >= 0, not only n >= 1.

In short, I see nothing wrong here.
>

Well, there could be two sources of confusion:
The Dedekind numbers oeis.org/A000372 consider antichains *of* subsets of
{1, 2, ..., n},
while oeis.org/A132581 considers antichains *in* the "first n elements of
the Boolean lattice",
where these n elements are identified with the first n nonnegative integers
{0, ..., n-1},
[ so it is confusing although not wrong that the comments consider
{0,...,n} ]
where an integer m = sum b_j 2^j  represents the set B(m) = { j >= 0 | b_j
= 1 }, subset of {0, 1, 2, ... }.
So,  A132581(2^k) counts all antichains whose *elements* are the sets B(0),
..., B(2^k-1),
which are the 2^k subsets of {0, ..., k-1}.
This is the same number as  A000372(k).

But JM Aranda did well to draw our attention to the comment in A132582,
although Sean is right by saying

> if you encounter a mistake, then you can simply edit the sequence and
> propose the correction.
>

Indeed, the first differences A132582(n) :=  A132581(n+1) - A132581(n)
count the antichains counted in  A132581(n+1) but not in A132581(n).
These are of course exactly those which contain the subset B(n) (not B(n+1)
!).
So the comment in  A132582 should say "containing n" (meaning "containing
B(n)"),
not "containing n-1" as written now (but OTOH, it is valid for all n >= 0)
-- where of course B(n) does not contain n, since the first one which does
contain n is B(2^n)!

So that comment was wrong, at least for the offsets as they stand.
But everything else is correct, although maybe slightly confusing at the
first glance.
I propose some hopefully clarifying comments / edits / examples in the two
sequences.

- Maximilian


> On Tue, 1 Jun 2021 at 01:49, jmmaranda wrote:
> > Dear Seqfans:
> > The EIS A132582 sequence must not have the term 0.
> > The published sequence is all displaced.
> > I have correctly calculated to term 204 but the bfile is rejected.
> > This mistake must be corrected.
> > How do we do it ?
> > I have created new Sourceforge projects, the link is:
> >
> > sourceforge.net/projects/eis-a132581/files
> > sourceforge.net/projects/eis-a132582/files
> >
> > JM Aranda
> > Madrid
> >
> > -------- A132582 COMMENTS
> > For n >= 1, <<<===!!!
> > a(n) is the number of antichains """containing (n-1) """ <<<===!!!
> >
> > in the first n elements of the infinite Boolean lattice.
> > Robert Israel, Mar 08 2017
> > ---------------------------------------------------------
> > CROSSREFS See A132581
> > A132582 as a simple table
> > n a(n)
> > 0 1 <<<===???
> > 1 1 2 2 3 1
> > 4 5 <<<---
> > 5 3 6 5 7 1
> > 8 19 <<<---
> > 9 14 10 25 14 19 15 1
> > 16 167 <<<---
> > 17 148 18 282 29 148 30 167 31 1
> > 32 7580 <<<---
> > 33 7413
> >
> > A132581 as a simple table
> > n a(n)
> > 0 1
> > 1 2
> > 2 3 <<<---
> > 3 5
> > 4 6 <<<---
>