OFFSET
0,5
LINKS
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.
FORMULA
From Petros Hadjicostas, Oct 10 2019: (Start)
T(n,0) = 1 for n >= 0.
T(n,1) = 2^n - 1 for n >= 1. [Dukes (2004), Theorem 2.1 (ii).]
T(n,k) = Sum_{m = k..n} binomial(n,m) * A058711(m,k) for n >= k. [Dukes (2004), see the equations before Theorem 2.1.]
(End)
EXAMPLE
Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
1;
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 36, 15, 1;
1, 31, 171, 171, 31, 1;
1, 63, 813, 2053, 813, 63, 1;
1, 127, 4012, 33442, 33442, 4012, 127, 1;
1, 255, 20891, 1022217, 8520812, 1022217, 20891, 255, 1;
...
CROSSREFS
Row sums give A058673.
KEYWORD
AUTHOR
N. J. A. Sloane, Dec 30 2000
STATUS
approved