Categories

[Jonathan Post]

Trapped on a holiday morning with no PC or even hand calculator about, I
started enumerating by hand the nonisomorphic partial endofunctions on n
indistinguishable objects.  That is, charter by digraphs with outdegree 0 or
2 at each vertex.  These are digraphs associated with partial functions,
becuase of the vertices with outdegree 0, as opposed to functional graphs
*digraphs where every vertex has outdegree 1).

By hand, subject to error, we have, for instance, 1 PFG (partial Functional
Graph) on 0 vertices, namely the null graph.  There are 2 PFGs on 1 vertex
(it either has or does not have a loop = 1-cycle).  There are 5 PFGs on 2
points: 0 arcs and 0, 1, or 2 loops on two disconnected vertices, and 1->2
where 2 either has no out-link or a loop.

For n=0,1,2,3,4,5 I count a(n) = 1, 22, 5, 16, 45, 121.

This is not in OEIS.  But I may have erred in my drawings, although they
still look good in double-checking.

Surely it is easy to enumerate these graphs in Mathematica, or PARI, or
Maple?  Although determining isomorphism can be slow when n gets big.

This is at least related to the Category enumeration, I think.

Happy Hanukkah, Merry Christmas, etc.,

Jonathan Vos Post

On 11/30/06, Henry Gould <gould at math.wvu.edu> wrote:
>
> I recall such a paper. The author published a paper giving a new
> definition  for the dimension of a topological space. He built up a
> chain of lemmas, ending up with a grand and glorious theorem based on
> the properties of a sequence of sets. The paper got bu the referees and
> the editors of a leading American research journal. When the
> Mathematical Reviews came out the very intelligent reviewer reported all
> the things in the paper and then ended with an incisive comment that
> unfortunately the author had failed to notice that every set in his
> sequence happened to be EMPTY. That was such a remarkable review.
>
> A friend and colleague  of mine made up a new definition of a pseudo
> inverse of  a matrix and was about to publuish it when it suddenly
> downed on him that none of the matrices he had invented existed.
>
> So, yes, this kind of thing can happen.
>
> Henry Gould
>
> = = = = = = =
>
> franktaw at netscape.net wrote:
> > I'd like to put my plug in for more application in math.
> >
> > I heard of a case some years ago where a paper was published about
> > some kind of mathematical object (I think it was a kind of topological
> > space, but it really doesn't matter).  Three or four more papers were
> > published, establishing more and more properties for this type of
> > object - until finally it was proved that they don't exist!  This
> > wouldn't have happened if somebody had asked for an example at an
> > early stage.
> >
> > There is a strong tendency in mathematics to start at the end.  The
> > researcher pursues a line of thought, which eventually leads to a
> > spiffy proof.  The proof is then published, with no hint of the
> > process by which it was reached.  This is a disservice to anybody who
> > might use a similar approach to solve some other problem.  It is
> > especially a disservice when presented to students.
> >
> > On a more personal level, I find when looking a math paper, I want to
> > know how this relates to problems that I am already interested in or
> > at least familiar with.  If I can't get an answer to that, I have a
> > hard time maintaining any interest.
> >
> > Franklin T. Adams-Watters
> >
> > ________________________________________________________________________
> > Check Out the new free AIM(R) Mail -- 2 GB of storage and
> > industry-leading spam and email virus protection.
> >
> >
>
>
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