Categories

[franktaw at netscape.net]

Thanks, Christian.

I think we need to look at one more table:  the number of connected
categories with n morphisms and k objects.  This starts:

  1
  2
  7 1
 35 6
228 28 2

(Row n has length ceiling(n/2).)

The table of the number of categories that I started with is a two-
dimensional Euler transform of this table.

The downward-sloping diagonal sums of this table are 1,3,15,???.
Shifting this left and taking the Euler transform gives the limiting
values Christian is referring to; starting with b(0):

1,3,21,???

For the table above, a(2n-1,n) has a purely algebraic or graph-
theoretic interpretation.  It is the number of connected
anti-transitive relations on n objects (meaning that if a R b and
b R c, then NOT (a R c)); equivalently, the number of bipartite
oriented trees, where each edge origin is in the same part.

If I haven't made any mistakes, this sequence starts:

1,1,2,3,6,10

This is not enough data to determine whether it is in the OEIS.

Franklin T. Adams-Watters


-----Original Message-----
From: bowerc at usa.net

I have one more row to add

------ Original Message ------

> 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.

________________________________________________________________________
Check Out the new free AIM(R) Mail -- 2 GB of storage and 
industry-leading spam and email virus protection.