Divisibility and Binomial Coefficients

[drew at math.mit.edu]

Dear SeqFans,

I have just contributed six sequences counting finite groupoids
(categories all of whose morphisms are invertible):
- A140185 (connected groupoids with n morphisms)
- A140186 (connected groupoids with n more morphisms than objects)
- A140187 (connected groupoids with n times as many morphisms as objects)
- A140188 (groupoids with n morphisms and k objects)
- A140189 (groupoids with n morphisms)
- A140190 (groupoids with m morphisms and (m-n) objects for any m>=2n)
The difficulty of counting groupoids is almost entirely in the
counting of groups, so that these sequences are not hard to compute
from A000001.

Maybe this is worth a new entry in the OEIS index ? (and by the way,
another entry for categories could be added):
categories: A125696, A125697, A125701
categories, connected: A125698, A125699, A125700, A125702
categories, strongly connected: [sequences to be added]
groupoids (categories with inverses): A140188, A140189, A140190
groupoids (categories with inverses), connected: A140185, A140186, A140187

For the curious, here are two surveys on groupoids:
- Ronald Brown, From groups to groupoids: a brief survey
http://www.bangor.ac.uk/~mas010/groupoidsurvey.pdf
- Alan Weinstein, Groupoids: unifying internal and external symmetry
http://www.ams.org/notices/199607/weinstein.pdf

Sincerely,
Benoit Jubin
[my oeis page: math.berkeley.edu/~jubin/oeis.html ]