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 A058710 Triangle T(n,k) giving number of loopless matroids of rank k on n labeled points (n >= 0, 0 <= k <= n). 8
 1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 14, 11, 1, 0, 1, 51, 106, 26, 1, 0, 1, 202, 1232, 642, 57, 1, 0, 1, 876, 22172, 28367, 3592, 120, 1, 0, 1, 4139, 803583, 8274374, 991829, 19903, 247, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS From Petros Hadjicostas, Oct 10 2019: (Start) The old references have some typos, some of which were corrected in the recent references (in 2004). Few additional typos were corrected here from the recent references. Here are some of the changes: T(5,2) = 31 --> 51 (see the comment by Ralf Stephan below); T(5,4) = 21 --> 26; sum of row n=5 is 185 (not 160 or 165); T(8,3) = 686515 --> 803583; T(8, 6) = 19904 --> 19903, and some others. This triangular array is the same as A058711 except that the current one has row n = 0 and column k = 0. (End) 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) = 0^n for n >= 0. T(n,1) = 1 for n >= 1. T(n,2) = Bell(n) - 1 = A000110(n) - 1 = A058692(n) for n >= 2. T(n,3) = Sum_{i = 3..n} Stirling2(n,i) * (A056642(i) - 1) = Sum_{i = 3..n} A008277(n,i) * A058720(i,3) for n >= 3. T(n,k) = Sum_{i = k..n} Stirling2(n,i) * A058720(i,k) for n >= k. [Dukes (2004), p. 3; see the equation with the Stirling numbers of the second kind.] (End) EXAMPLE Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:   1;   0, 1;   0, 1,    1;   0, 1,    4,      1;   0, 1,   14,     11,       1;   0, 1,   51,    106,      26,      1;   0, 1,  202,   1232,     642,     57,     1;   0, 1,  876,  22172,   28367,   3592,   120,   1;   0, 1, 4139, 803583, 8274374, 991829, 19903, 247, 1;   ... CROSSREFS Cf. Same as A058711 (except for row n=0 and column k=0). Row sums give A058712. Columns include (truncated versions of) A000007 (k=0), A000012 (k=1), A058692 (k=2), A058715 (k=3). Sequence in context: A055105 A200545 A294522 * A281891 A124539 A249094 Adjacent sequences:  A058707 A058708 A058709 * A058711 A058712 A058713 KEYWORD nonn,tabl,nice AUTHOR N. J. A. Sloane, Dec 31 2000 EXTENSIONS T(5,2) corrected from 31 to 51 by Ralf Stephan, Nov 29 2004 STATUS approved

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Last modified February 20 13:20 EST 2020. Contains 332077 sequences. (Running on oeis4.)