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

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

1,5

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

Table of n, a(n) for n=1..45.

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

nLab, Cauchy complete category.

nLab, Karoubi envelope.

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.

Example

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

Crossrefs

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

Sequence in context: A133644 A265870 A341409 * A284466 A258615 A152431

Adjacent sequences: A384131 A384132 A384133 * A384135 A384136 A384137

Keyword

nonn,tabl,hard,more

Author

Elijah Beregovsky, May 20 2025

Status

approved