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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
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, 383, 185981, 188936, 1092, 66, 7, 1, 1, 5249, 4884573865 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,5

COMMENTS

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

LINKS

Table of n, a(n) for n=2..40.

Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory. II. Combinatorial geometries, Studies in Appl. Math. 49 (1970), 109-133. [Annotated scanned copy of pages 126 and 127 only]

Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory. II. Combinatorial geometries, Studies in Appl. Math. 49 (1970), 109-133.

W. M. B. Dukes, Tables of matroids.

W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.

W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.

W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.

Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007. [See Table 2, p. 9.]

Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431. [See Table 2, p. 420.]

Y. Matsumoto, S. Moriyama, H. Imai, and D. Bremmer, Matroid enumeration for incidence geometry, Discrete Comput. Geom. 47 (2012), 17-43.

Gordon Royle and Dillon Mayhew, 9-element matroids.

Index entries for sequences related to matroids

FORMULA

From Petros Hadjicostas, Oct 09 2019: (Start)

T(n, n-1) = n-2 for n >= 2. [Dukes (2004), Lemma 2.2(ii).]

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

(End)

EXAMPLE

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

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

  ...

From Petros Hadjicostas, Oct 09 2019: (Start)

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

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

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

(End)

CROSSREFS

Cf. A058720. Row sums give A002773.

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

Sequence in context: A125781 A091150 A091351 * A112705 A070895 A127054

Adjacent sequences:  A058727 A058728 A058729 * A058731 A058732 A058733

KEYWORD

nonn,tabl,nice,more

AUTHOR

N. J. A. Sloane, Dec 31 2000

EXTENSIONS

Row n=9 from Petros Hadjicostas, Oct 09 2019 using the papers by Mayhew and Royle

STATUS

approved

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Last modified February 27 10:15 EST 2020. Contains 332304 sequences. (Running on oeis4.)