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 A058730 Triangle T(n,k) giving number of nonisomorphic simple matroids of rank k on n labeled points (n >= 2, 2 <= k <= n). 4

%I

%S 1,1,1,1,2,1,1,4,3,1,1,9,11,4,1,1,23,49,22,5,1,1,68,617,217,40,6,1,1,

%T 383,185981,188936,1092,66,7,1,1,5249,4884573865

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

%C To make this sequence a triangular array, we assume n >= 2 and 2 <= k <= n. According to the references, however, we have T(0,0) = T(1, 1) = 1, and 0 in all other cases. - _Petros Hadjicostas_, Oct 09 2019

%H Henry H. Crapo and Gian-Carlo Rota, <a href="/A002773/a002773.pdf">On the foundations of combinatorial theory. II. Combinatorial geometries</a>, Studies in Appl. Math. 49 (1970), 109-133. [Annotated scanned copy of pages 126 and 127 only]

%H Henry H. Crapo and Gian-Carlo Rota, <a href="https://doi.org/10.1002/sapm1970492109">On the foundations of combinatorial theory. II. Combinatorial geometries</a>, Studies in Appl. Math. 49 (1970), 109-133.

%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>, SÃ©minaire Lotharingien de Combinatoire 51 (2004), Article B51g.

%H Dillon Mayhew and Gordon F. Royle, <a href="https://arxiv.org/abs/math/0702316">Matroids with nine elements</a>, arXiv:math/0702316 [math.CO], 2007. [See Table 2, p. 9.]

%H Dillon Mayhew and Gordon F. Royle, <a href="https://doi.org/10.1016/j.jctb.2007.07.005">Matroids with nine elements</a>, J. Combin. Theory Ser. B 98(2) (2008), 415-431. [See Table 2, p. 420.]

%H Y. Matsumoto, S. Moriyama, H. Imai, and D. Bremmer, <a href="https://doi.org/10.1007/s00454-011-9388-y">Matroid enumeration for incidence geometry</a>, Discrete Comput. Geom. 47 (2012), 17-43.

%H Gordon Royle and Dillon Mayhew, <a href="https://web.archive.org/web/20080828102733/http://people.csse.uwa.edu.au/gordon/matroid-integer-sequences.html">9-element matroids</a>.

%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) = n-2 for n >= 2. [Dukes (2004), Lemma 2.2(ii).]

%F T(n, n-2) = 6 - 4*n + Sum_{k = 1..n} A000041(k) for n >= 3. [Dukes (2004), Lemma 2.2(iv).]

%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, 2, 1;

%e 1, 4, 3, 1;

%e 1, 9, 11, 4, 1;

%e 1, 23, 49, 22, 5, 1;

%e 1, 68, 617, 217, 40, 6, 1;

%e 1, 383, 185981, 188936, 1092, 66, 7, 1;

%e ...

%e From _Petros Hadjicostas_, Oct 09 2019: (Start)

%e Matsumoto et al. (2012, p. 36) gave an incomplete row n = 10 (starting at k = 2):

%e 1, 5249, 4884573865, *, 4886374072, 9742, 104, 8, 1;

%e They also gave incomplete rows for n = 11 and n = 12.

%e (End)

%Y Cf. A058720. Row sums give A002773.

%Y Columns include (truncations of) A000012 (k=2), A058731 (k=3), A058733 (k=4).

%K nonn,tabl,nice,more

%O 2,5

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

%E Row n=9 from _Petros Hadjicostas_, Oct 09 2019 using the papers by Mayhew and Royle

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Last modified April 1 05:04 EDT 2020. Contains 333155 sequences. (Running on oeis4.)