cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A125701 Limiting values for category table A125697.

Original entry on oeis.org

1, 3, 21, 132, 950, 7698
Offset: 0

Views

Author

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

Keywords

Comments

Euler transform of A125700 (ignoring the initial 1).

Links

Crossrefs

Cf. A125697, A125700, A125696.

Formula

G.f.: Product_{i>=1} 1/(1-x^i)^A125700(i).
A125697(n,k) = a(n-k) if k >= (2/3)*n.

Extensions

a(4) from Ben Spitz, Jul 18 2023
a(5) from Ben Spitz, Aug 30 2023

A125696 Number of categories with n morphisms.

Original entry on oeis.org

1, 1, 3, 11, 55, 329, 2858, 36440, 1723286, 3687822810
Offset: 0

Views

Author

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

Keywords

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

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

A384066 Limiting values for Cauchy-complete category table A384134.

Original entry on oeis.org

1, 2, 10, 39, 168
Offset: 0

Views

Author

Elijah Beregovsky, May 20 2025

Keywords

Comments

This appears to be the absolute value of A165814.

Crossrefs

Cf. A384134, A125697, A125701.

Formula

A384134(n,k) = a(n-k) if k >= (2/3)*n.

A384134 Triangle read by rows: T(n,k) is the number of Cauchy-complete categories with n morphisms and k objects.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 6, 2, 1, 1, 12, 9, 2, 1, 2, 23, 25, 10, 2, 1, 1, 45, 69, 35, 10, 2, 1, 5, 98, 178, 119, 38, 10, 2, 1, 2, 278, 457, 371, 151, 39, 10, 2, 1
Offset: 1

Views

Author

Elijah Beregovsky, May 20 2025

Keywords

Comments

A Cauchy complete (also called Karoubi complete or idempotent-complete) category is one in which all idempotents split. In other words, in a Cauchy-complete category every arrow e:A->A such that e=e*e has a retract, meaning there exists an object B and morphisms r:A→B and s:B→A such that s∘r=e but r∘s=1_B.

Examples

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  2,  6,  2,  1;
  1, 12,  9,  2, 1;
  2, 23, 25, 10, 2, 1;
  ...

Links

Crossrefs

Cf. A384135 (row sums), A000001 (column 1), A384066 (limiting values), A125697.

Formula

T(n,k) = A384066(n-k) if k >= (2/3)*n.
T(3n,2n) = T(3n-1,2n-1) + 1 when n >= 1.
T(3n-1,2n-1) = T(3n-2,2n-2) + 3 when n >= 2.
T(3n-2,2n-2) = T(3n-3,2n-3) + 13 when n >= 4.

A384135 Number of Cauchy-complete categories with n morphisms.

Original entry on oeis.org

1, 2, 4, 11, 25, 63, 163, 451, 1311
Offset: 1

Views

Author

Elijah Beregovsky, May 20 2025

Keywords

Comments

A Cauchy-complete (also called Karoubi-complete or idempotent-complete) category is one in which all idempotents split. In other words, in a Cauchy-complete category every arrow e:A→A such that e=e∘e has a retract, meaning there exists an object B and morphisms r:A→B and s:B→A such that s∘r=e but r∘s=1_B.

Links

Crossrefs

Cf. A125697.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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