A055080 Triangle T(n,k) read by rows, giving number of k-member minimal covers of an unlabeled n-set, k=1..n.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 9, 4, 1, 1, 9, 23, 17, 5, 1, 1, 12, 51, 65, 28, 6, 1, 1, 16, 103, 230, 156, 43, 7, 1, 1, 20, 196, 736, 863, 336, 62, 8, 1, 1, 25, 348, 2197, 4571, 2864, 664, 86, 9, 1, 1, 30, 590, 6093, 22952, 25326, 8609, 1229, 115, 10, 1, 1, 36, 960

Offset: 1

Author

Vladeta Jovovic, Jun 13 2000

Keywords

• nonn
• tabl

Comments

Also number of unlabeled split graphs on n vertices and with a k-element clique (cf. A048194).

Examples

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   3,   1;
  1,  6,   9,   4,   1;
  1,  9,  23,  17,   5,   1;
  1, 12,  51,  65,  28,   6,  1;
  1, 16, 103, 230, 156,  43,  7, 1;
  1, 20, 196, 736, 863, 336, 62, 8, 1;
  ...
There are four minimal covers of an unlabeled 3-set: one 1-cover {{1,2,3}}, two 2-covers {{1,2},{3}}, {{1,2},{1,3}} and one 3-cover {{1},{2},{3}}.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
• R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
• Vladeta Jovovic, Binary matrices up to row and column permutations
• G. F. Royle, Counting Set Covers and Split Graphs, J. Integer Seqs., 3 (2000), #00.2.6.
• Eric Weisstein's World of Mathematics, Minimal covers

Crossrefs

Row sums give A048194.

Columns are A000012, A087811, A005783, A005784, A005785 A005786, A055066.

Diagonals are A000012, A000027, A005744, A005745, A005746, A005771, A005747, A005748.

Cf. A035348 for labeled case.

Cf. A002620, A028657.

Programs

PARI

\\ Needs A(n,m) from A028657.
T(n,k) = A(n-k, k) - if(kAndrew Howroyd, Feb 28 2023

Formula

T(n,k) = A028657(n,k) - A028657(n-1,k). - Andrew Howroyd, Feb 28 2023