A058720 Triangle T(n,k) giving the number of simple matroids of rank k on n labeled points (n >= 2, 2 <= k <= n).

1, 1, 1, 1, 5, 1, 1, 31, 16, 1, 1, 352, 337, 42, 1, 1, 8389, 18700, 2570, 99, 1, 1, 433038, 7642631, 907647, 16865, 219, 1

Offset: 2

Author

N. J. A. Sloane, Dec 31 2000

Keywords

• nonn
• tabl
• nice
• more

Examples

			Triangle T(n,k) (with rows n >= 2 and columns k >= 2) begins as follows:
  1;
  1,      1;
  1,      5,       1;
  1,     31,      16,      1;
  1,    352,     337,     42,     1;
  1,   8389,   18700,   2570,    99,   1;
  1, 433038, 7642631, 907647, 16865, 219, 1;
  ...
		

Links

• Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, and Martin Leuner, On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture, arXiv:1907.01073 [math.CO], 2019.
• W. M. B. Dukes, Tables of matroids.
• W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
• W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
• W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g. [See p. 11.]
• Index entries for sequences related to matroids

Crossrefs

Cf. A000110 (Bell numbers), A002662, A058710, A058711, A058716, A058730, A100728.

Row sums give A058721.

Columns include (truncated versions of) A000012 (k=2), (A056642)+1 (k=3), A058722 (k=4).

Formula

From Petros Hadjicostas, Oct 09 2019: (Start)

T(n, n-1) = 2^n - 1 - binomial(n+1,2) = A002662(n) for n >= 2. [Dukes (2004), Lemma 2.2(i).]

T(n, n-2) = A100728(n) = A000110(n+1) + binomial(n+3,4) + 2*binomial(n+1,4) - 2^n - 2^(n-1)*binomial(n+1,2). [Dukes (2004), Lemma 2.2(iii).]

(End)