cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A001200 Number of linear geometries on n (unlabeled) points.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 10, 24, 69, 384, 5250, 232929, 28872973
Offset: 0

Views

Author

N. J. A. Sloane, D.Glynn(AT)math.canterbury.ac.nz

Keywords

Comments

For the labeled case see A056642.
Also a(n) = 1 + number of non-isomorphic simple rank-3 matroids on n elements (see A058731); a(n) = number of non-isomorphic 2-partitions of a set of size n. For 1-partitions see A000041.

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #42.
  • CRC Handbook of Combinatorial Designs, 1996, pp. 216, 697.
  • J. Doyen, Sur le nombre d'espaces linéaires non isomorphes de n points. Bull. Soc. Math. Belg. 19 1967 421-437.
  • P. Robillard, On the weighted finite linear spaces. Bull. Soc. Math. Belg. 22 (1970), 227-241.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

A056642 Number of linear spaces on n (labeled) points.

Original entry on oeis.org

1, 1, 2, 6, 32, 353, 8390, 433039, 50166354, 13480967630
Offset: 1

Views

Author

W. M. B. Dukes (dukes(AT)stp.dias.ie), Aug 28 2000

Keywords

Comments

Alternatively, number of linear geometries on n (labeled) points. For the unlabeled case see A001200.
Also a(n) = 1 + number of simple rank-3 matroids on n (labeled) elements; a(n) = number of 2-partitions of a set of size n.

References

  • L. M. Batten and A. Beutelspacher: The theory of finite linear spaces, Cambridge Univ. Press, 1993 (see the Appendix).
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #42.
  • J. Doyen, Sur le nombre d'espaces linéaires non isomorphes de n points, Bull. Soc. Math. Belg. 19 (1967), 421-437.
  • J. A. Thas, Sur le nombre d'espaces linéaires non isomorphes de n points, Bull. Soc. Math. Belg. 21 (1969), 57-66.

Crossrefs

Corrected version of A001199. Cf. A002773, A001200, A031436, A058731.

Extensions

a(9) and a(10) from Gordon Royle, May 29 2006

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

Original entry on oeis.org

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
Offset: 2

Views

Author

N. J. A. Sloane, Dec 31 2000

Keywords

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

Examples

			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).

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)

Extensions

Row n=9 from Petros Hadjicostas, Oct 09 2019 using the papers by Mayhew and Royle
Showing 1-3 of 3 results.