A125696 Number of categories with n morphisms.

1, 1, 3, 11, 55, 329, 2858, 36440, 1723286, 3687822810

Offset: 0

Author

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

Keywords

• hard
• more
• nonn

Examples

			The 11 categories with 3 morphisms consist of:
* 7=A058129(3) categories with 1 object (monoids),
* 3 categories with 2 objects, consisting of: 2=A058129(2) disconnected combinations of a 2-element monoid and a 1-element monoid, and the category with 2 objects and a single morphism between the two objects,
* 1 category with 3 objects (3 separate 1-element monoids).
		

Links

• Geoff Cruttwell, Counting Finite Categories, presentation, (2018).
• Eric Weisstein's World of Mathematics, Category

Crossrefs

Cf. A058129, A125697, A125698, A125720.

Formula

Euler transform of A125698.

G.f.: Product_{i>=1} 1/(1-x^i)^A125698(i).

From Elijah Beregovsky, May 13 2025 (Start)

a(n) >= A058129(n).

Conjecture: a(n) = A058129(n)*(1+o(n)). See Cruttwell presentation in Links. (End)

Extensions

a(0) and a(7)-a(9) from Thomas Anton, from the work of G. Cruttwell and R. Leblanc, Jan 25 2019