A125698 -id:A125698 - cp's OEIS frontend

A125696 Number of categories with n morphisms.

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

Offset: 0

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

			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).

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

Cf. A058129, A125697, A125698, A125720.

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)

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

A125699 Table, T(n,k) is the number of connected categories with n morphisms and k objects.

Original entry on oeis.org

1, 2, 7, 1, 35, 6, 228, 28, 2, 2237, 159, 11, 31559, 1075, 77, 3, 1668997, 9389, 497, 24

Offset: 1

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

Connected in the sense that if the morphism direction and composition is ignored, resulting in a multigraph, that multigraph is connected.

			The table starts:
        1;
        2;
        7,    1;
       35,    6;
      228,   28,   2;
     2237,  159,  11;
    31559, 1075,  77,  3;
  1668997, 9389, 497, 24;
  ...

Ben Spitz, SmallCategories
Eric Weisstein's World of Mathematics, Category

Cf. A125696, A125698 (row sums), A110654 (row lengths), A058129 (column 1), A125700 (diagonal sums), A125702 (T(2n-1, n)).

a(14)-a(20) from Ben Spitz, Sep 02 2023

cp's OEIS Frontend

A125696 Number of categories with n morphisms.

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