Categories

[Christian G. Bower]

I have row 6 now:

> 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
2237 485 111 21 3 1

> Taking the row sums, we get:
>
> 1,3,11,55
329
2858

>  The
> inverse Euler
> transform,
>
> 1,2,8,41
258
2407

> 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
2237 159 11
31559 ?   ? 3

> (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,???.
77
> Shifting this left and taking the Euler transform gives the limiting
> values Christian is referring to; starting with b(0):
> 
> 1,3,21,???
132