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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). 7

%I

%S 1,1,1,1,5,1,1,31,16,1,1,352,337,42,1,1,8389,18700,2570,99,1,1,433038,

%T 7642631,907647,16865,219,1

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

%H 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 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. [See p. 11.]

%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, n-1) = 2^n - 1 - binomial(n+1,2) = A002662(n) for n >= 2. [Dukes (2004), Lemma 2.2(i).]

%F 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 (End)

%e Triangle T(n,k) (with rows n >= 2 and columns k >= 2) begins as follows:

%e 1;

%e 1, 1;

%e 1, 5, 1;

%e 1, 31, 16, 1;

%e 1, 352, 337, 42, 1;

%e 1, 8389, 18700, 2570, 99, 1;

%e 1, 433038, 7642631, 907647, 16865, 219, 1;

%e ...

%Y Cf. A000110 (Bell numbers), A002662, A058710, A058711, A058716, A058730, A100728.

%Y Row sums give A058721.

%Y Columns include (truncated versions of) A000012 (k=2), (A056642)+1 (k=3), A058722 (k=4).

%K nonn,tabl,nice,more

%O 2,5

%A _N. J. A. Sloane_, Dec 31 2000

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Last modified February 24 09:40 EST 2020. Contains 332209 sequences. (Running on oeis4.)