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%I #27 Oct 11 2019 06:31:37
%S 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,
%T 63,1,1,127,4012,33442,33442,4012,127,1,1,255,20891,1022217,8520812,
%U 1022217,20891,255,1,1,511
%N Triangle T(n,k) read by rows, giving number of matroids of rank k on n labeled points (n >= 0, 0 <= k <= n).
%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 10 2019: (Start)
%F T(n,0) = 1 for n >= 0.
%F T(n,1) = 2^n - 1 for n >= 1. [Dukes (2004), Theorem 2.1 (ii).
%F T(n,2) = Bell(n+1) - 2^n = A000110(n+1) - A000079(n) for n >= 2. [Dukes (2004), Theorem 2.1 (ii).]
%F 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.]
%F (End)
%e Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 7, 7, 1;
%e 1, 15, 36, 15, 1;
%e 1, 31, 171, 171, 31, 1;
%e 1, 63, 813, 2053, 813, 63, 1;
%e 1, 127, 4012, 33442, 33442, 4012, 127, 1;
%e 1, 255, 20891, 1022217, 8520812, 1022217, 20891, 255, 1;
%e ...
%Y Row sums give A058673.
%Y Columns include (truncated versions of) A000012 (k=0), A000225 (k=1), A058681 (k=2), A058687 (k=3).
%Y Cf. A000079, A000110, A053534, A058710, A058711.
%K nonn,nice,tabl,more
%O 0,5
%A _N. J. A. Sloane_, Dec 30 2000
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