A125697 -id:A125697

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A125701

Limiting values for category table A125697.

+20
2

1, 3, 21, 132, 950, 7698

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Euler transform of A125700 (ignoring the initial 1).

LINKS

Table of n, a(n) for n=0..5.

Ben Spitz, SmallCategories

FORMULA

G.f.: Product_{i>=1} 1/(1-x^i)^A125700(i).

A125697(n,k) = a(n-k) if k >= (2/3)*n.

CROSSREFS

Cf. A125697, A125700, A125696.

KEYWORD

nonn,hard,more

AUTHOR

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

EXTENSIONS

a(4) from Ben Spitz, Jul 18 2023

a(5) from Ben Spitz, Aug 30 2023

STATUS

approved

A125696

Number of categories with n morphisms.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..9.

Geoff Cruttwell, Counting Finite Categories, presentation, (2018).

Eric Weisstein's World of Mathematics, Category

FORMULA

Euler transform of A125698.

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

EXAMPLE

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).

CROSSREFS

Cf. A058129, A125697, A125698, A125720.

KEYWORD

hard,more,nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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