A125699 -id:A125699 - cp's OEIS frontend

A125702 Number of connected categories with n objects and 2n-1 morphisms.

1, 1, 2, 3, 6, 10, 22, 42, 94, 203, 470, 1082, 2602, 6270, 15482, 38525, 97258, 247448, 635910, 1645411, 4289010, 11245670, 29656148, 78595028, 209273780, 559574414, 1502130920, 4046853091, 10939133170, 29661655793

Offset: 1

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

nonn

Also number of connected antitransitive relations on n objects (antitransitive meaning a R b and b R c implies not a R c); equivalently, number of free oriented bipartite trees, with all arrows going from one part to the other part.

Also the number of non-isomorphic multi-hypertrees of weight n - 1 with singletons allowed. A multi-hypertree with singletons allowed is a connected set multipartition (multiset of sets) with density -1, where the density of a set multipartition is the weight (sum of sizes of the parts) minus the number of parts minus the number of vertices. - Gus Wiseman, Oct 30 2018

			From _Gus Wiseman_, Oct 30 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(6) = 10 multi-hypertrees of weight n - 1 with singletons allowed:
  {}  {{1}}  {{12}}    {{123}}      {{1234}}        {{12345}}
             {{1}{1}}  {{2}{12}}    {{13}{23}}      {{14}{234}}
                       {{1}{1}{1}}  {{3}{123}}      {{4}{1234}}
                                    {{1}{2}{12}}    {{2}{13}{23}}
                                    {{2}{2}{12}}    {{2}{3}{123}}
                                    {{1}{1}{1}{1}}  {{3}{13}{23}}
                                                    {{3}{3}{123}}
                                                    {{1}{2}{2}{12}}
                                                    {{2}{2}{2}{12}}
                                                    {{1}{1}{1}{1}{1}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..500

Same as A122086 except for n = 1; see there for formulas. Cf. A125699.

Cf. A000081, A000272, A007716, A007717, A030019, A052888, A134954, A317631, A317632, A318697, A320921, A321155.

\\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={Vec(2*TreeGf(n) - TreeGf(n)^2 - x)} \\ Andrew Howroyd, Nov 02 2019

a(n) = A122086(n) for n > 1.

G.f.: 2*f(x) - f(x)^2 - x where f(x) is the g.f. of A000081. - Andrew Howroyd, Nov 02 2019

A125697 Table, T(n,k) is the number of categories with n morphisms and k objects.

Original entry on oeis.org

1, 2, 1, 7, 3, 1, 35, 16, 3, 1, 228, 77, 20, 3, 1, 2237, 485, 111, 21, 3, 1, 31559, 4013, 716, 127, 21, 3, 1, 1668997, 47648, 5623, 862, 131, 21, 3, 1, 3685886630, 1868157, 60201, 6739, 926, 132, 21, 3, 1

Offset: 1

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

This is a two-dimensional Euler transform of A125699.

			The table starts:
     1;
     2,   1;
     7,   3,   1;
    35,  16,   3,  1;
   228,  77,  20,  3, 1;
  2237, 485, 111, 21, 3, 1;
  ...

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

Cf. A125696 (row sums), A058129 (column 1), A125699, A125701.

G.f.: Product_{i>=1} Product_{j=1..ceiling(i/2)} 1/(1 - x^i y^j)^A125699(i,j).
T(n,k) = A125701(n-k) when k >= (2/3)*n.
From Ben Spitz, Aug 30 2023: (Start)
T(3n,2n) = T(3n-1,2n-1) + 1 when n >= 1.
T(3n-1,2n-1) = T(3n-2,2n-2) + 4 when n >= 2.
T(3n-2,2n-2) = T(3n-3,2n-3) + 19 when n >= 4.
(End)

a(23)-a(29) from Ben Spitz, Jul 17 2023
a(30)-a(36) from Ben Spitz, Aug 29 2023
a(37)-a(45) from Elijah Beregovsky, after the work of Cruttwell and Leblanc, May 20 2025

A125700 Number of connected categories with n more morphisms than objects.

Original entry on oeis.org

1, 3, 15, 77, 494, 4068

Offset: 0

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

Eric Weisstein's World of Mathematics, Category

Cf. A125696, A125699, A125701.

a(n) = Sum_{k=1..n+1} A125699(n+k,k).

a(4)-a(5) from Ben Spitz, Aug 30 2023

A125698 Number of connected categories with n morphisms.

Original entry on oeis.org

1, 2, 8, 41, 258, 2407

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.

Inverse Euler transform of A125696.

			The 8 categories with 3 morphisms consist of 7 3-element monoids and one 2-object category with a single morphism between the objects.

Eric Weisstein's World of Mathematics, Category

Cf. A125696, A125699, A125700.

cp's OEIS Frontend

A125702 Number of connected categories with n objects and 2n-1 morphisms.

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A125697 Table, T(n,k) is the number of categories with n morphisms and k objects.

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A125700 Number of connected categories with n more morphisms than objects.

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A125698 Number of connected categories with n morphisms.

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