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%I #38 Oct 10 2019 04:28:14
%S 1,1,1,1,4,1,1,14,11,1,1,51,106,26,1,1,202,1232,642,57,1,1,876,22172,
%T 28367,3592,120,1,1,4139,803583,8274374,991829,19903,247,1
%N Triangle T(n,k) giving the number of loopless matroids of rank k on n labeled points (n >= 1, 1 <= k <= n).
%C From _Petros Hadjicostas_, Oct 09 2019: (Start)
%C The old references had some typos, some of which were corrected in the recent ones. Few additional typos were corrected here from the recent references. Here are some of the changes: T(5,2) = 31 --> 51; 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.
%C This triangular array is the same as A058710 except that it has no row n = 0 and no column k = 0.
%C (End)
%H W. M. B. Dukes, <a href="http://www.stp.dias.ie/~dukes/matroid.html">Tables of matroids</a>.
%H W. M. B. Dukes, <a href="https://web.archive.org/web/20030208144026/http://www.stp.dias.ie/~dukes/phd.html">Counting and Probability in Matroid Theory</a>, Ph.D. Thesis, Trinity College, Dublin, 2000.
%H W. M. B. Dukes, <a href="https://arxiv.org/abs/math/0411557">The number of matroids on a finite set</a>, arXiv:math/0411557 [math.CO], 2004.
%H W. M. B. Dukes, <a href="http://emis.impa.br/EMIS/journals/SLC/wpapers/s51dukes.html">On the number of matroids on a finite set</a>, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
%H <a href="/index/Mat#matroid">Index entries for sequences related to matroids</a>
%F From _Petros Hadjicostas_, Oct 09 2019: (Start)
%F T(n,1) = 1 for n >= 1.
%F T(n,2) = Bell(n) - 1 = A000110(n) - 1 = A058692(n) for n >= 2.
%F 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.
%F 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.]
%F (End)
%e Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 14, 11, 1;
%e 1, 51, 106, 26, 1;
%e 1, 202, 1232, 642, 57, 1;
%e 1, 876, 22172, 28367, 3592, 120, 1;
%e 1, 4139, 803583, 8274374, 991829, 19903, 247, 1;
%e ...
%Y Same as A058710 (except for row n=0 and column k=0).
%Y Row sums give A058712.
%Y Columns include (truncated versions of) A000012 (k=1), A058692 (k=2), A058715 (k=3).
%K nonn,nice,tabl,more
%O 1,5
%A _N. J. A. Sloane_, Dec 31 2000
%E Several values corrected by _Petros Hadjicostas_, Oct 09 2019
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