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A058720 Triangle T(n,k) giving the number of simple matroids of rank k on n labeled points (n >= 2, 2 <= k <= n).

%I A058720 #30 Oct 10 2019 04:28:24
%S A058720 1,1,1,1,5,1,1,31,16,1,1,352,337,42,1,1,8389,18700,2570,99,1,1,433038,
%T A058720 7642631,907647,16865,219,1
%N A058720 Triangle T(n,k) giving the number of simple matroids of rank k on n labeled points (n >= 2, 2 <= k <= n).
%H A058720 Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, and Martin Leuner, <a href="https://arxiv.org/abs/1907.01073">On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture</a>, arXiv:1907.01073 [math.CO], 2019.
%H A058720 W. M. B. Dukes, <a href="http://www.stp.dias.ie/~dukes/matroid.html">Tables of matroids</a>.
%H A058720 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 A058720 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 A058720 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. [See p. 11.]
%H A058720 <a href="/index/Mat#matroid">Index entries for sequences related to matroids</a>
%F A058720 From _Petros Hadjicostas_, Oct 09 2019: (Start)
%F A058720 T(n, n-1) = 2^n - 1 - binomial(n+1,2) = A002662(n) for n >= 2. [Dukes (2004), Lemma 2.2(i).]
%F A058720 T(n, n-2) = A100728(n) = A000110(n+1) + binomial(n+3,4) + 2*binomial(n+1,4) - 2^n - 2^(n-1)*binomial(n+1,2). [Dukes (2004), Lemma 2.2(iii).]
%F A058720 (End)
%e A058720 Triangle T(n,k) (with rows n >= 2 and columns k >= 2) begins as follows:
%e A058720   1;
%e A058720   1,      1;
%e A058720   1,      5,       1;
%e A058720   1,     31,      16,      1;
%e A058720   1,    352,     337,     42,     1;
%e A058720   1,   8389,   18700,   2570,    99,   1;
%e A058720   1, 433038, 7642631, 907647, 16865, 219, 1;
%e A058720   ...
%Y A058720 Cf. A000110 (Bell numbers), A002662, A058710, A058711, A058716, A058730, A100728.
%Y A058720 Row sums give A058721.
%Y A058720 Columns include (truncated versions of) A000012 (k=2), (A056642)+1 (k=3), A058722 (k=4).
%K A058720 nonn,tabl,nice,more
%O A058720 2,5
%A A058720 _N. J. A. Sloane_, Dec 31 2000