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 A058669 Triangle T(n,k) read by rows, giving number of matroids of rank k on n labeled points (n >= 0, 0 <= k <= n). 5
 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, 1, 511 (list; table; graph; refs; listen; history; text; internal format)
 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, SÃ©minaire 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,2) = Bell(n+1) - 2^n = A000110(n+1) - A000079(n) for n >= 2. [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. Columns include (truncated versions of) A000012 (k=0), A000225 (k=1), A058681 (k=2), A058687 (k=3). Cf. A000079, A000110, A053534, A058710, A058711. Sequence in context: A184173 A022166 A141689 * A057004 A059328 A174387 Adjacent sequences:  A058666 A058667 A058668 * A058670 A058671 A058672 KEYWORD nonn,nice,tabl,more,changed AUTHOR N. J. A. Sloane, Dec 30 2000 STATUS approved

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Last modified October 23 14:22 EDT 2019. Contains 328345 sequences. (Running on oeis4.)