Categories

[Jonathan Post]

Thank you, Christian.  That looks right.

Note that the partial sums of the first column are A118601 Number of monoids
(semigroups with identity) of order <=n.  This first column also equals the
row sums of A058137, which partitions the monoids of order n by the number k
of idempotents.






On 11/29/06, Christian G. Bower <bowerc at usa.net> wrote:
>
>
> I have one more row to add
>
> ------ Original Message ------
> Received: Wed, 22 Nov 2006 01:21:01 PM PST
> From: franktaw at netscape.net
> To: seqfan at ext.jussieu.fr
> Subject: Categories
>
> > How many categories are there?
> >
> > First, how many categories are there with n morphisms and k objects?
> > This table starts:
> >
> >  1
> >  2  1
> >  7  3 1
> > 35 16 3 1
> >
> 228 77 20 3 1
>
> > The first column is A058129, the number of monoids; the main diagonal
> > is all 1's.  I am not
> > 100% certain of the 16 in the final row.
> >
> > Taking the row sums, we get:
> >
> > 1,3,11,55
> 329
> >
> > the number of categories with n morphisms.  This is probably not in the
> > OEIS (only
> > A001776 is possible - other matches become less than A058129).
>
> While the 329 is eerily close to A001776's 330, EULER(A058129) is a
> lower limit for this sequence and we have 2982 vs 2345 for #6 there.
> >  The
> > inverse Euler
> > transform,
> >
> > 1,2,8,41
> 258
> >
> > is the number of connected categories with n morphisms; this is
> > likewise probably not
> > in the OEIS (only A052447 is possible).
> No longer possible
> >
> > Can somebody generate more data?
> >
> > Franklin T. Adams-Watters
> >
>
> Christian
>
> PS
>
> I don't know if it will be feasible to collect enough data to do
> this, but the columns of the triangle converge and that convergence
> would make an interesting sequence in and of itself.
>
>
>
>
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