Categories

[Christian G. Bower]

I have one more row to add

------ Original Message ------
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.