A001548 Number of connected linear spaces with n (unlabeled) points.

1, 1, 0, 1, 1, 2, 4, 13, 42, 308, 4845, 227613, 28639650

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice
• more

Comments

Euler transform is A001200. - Michael Somos, Apr 24 2014

In any linear space any two distinct points belong to exactly one line. A linear space is disconnected if there exists a partition of the points of the space into two subsets such that for any two distinct points in a subset of the partition the unique line they both belong to is completely contained in that subset. - Michael Somos, Apr 24 2014

Examples

			a(2) = 0 because the unique linear space on two points can be partitioned into two single point subsets which disconnects the space vacuously. a(5) = 2 because there are two connected linear spaces with 5 points: one has only one line and the other has two lines with three points that intersect in one point that belongs to no other line while the other four points belong to three lines. - _Michael Somos_, Apr 24 2014
		

References

• L. M. Batten and A. Beutelspacher: The theory of finite linear spaces, Cambridge Univ. Press, 1993 (see the Appendix).
• Doyen, Jean; Sur le nombre d'espaces linéaires non isomorphes de n points. Bull. Soc. Math. Belg. 19 1967 421-437.
• 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).

Links

• J. Doyen, Sur le nombre d'espaces lineaires non isomorphes de n points [Annotated and scanned copy]
• Olaf Lechtenfeld, Konrad Schwerdtfeger and Johannes Thürigen, N=4 Multi-Particle Mechanics, WDVV Equation and Roots, SIGMA 7 (2011), 023. See the table on p. 18 for the sequence a(n)-1 for n > 2.
• Pierre Robillard, On the weighted finite linear spaces, Bull. Soc. Math. Belg. 22 (1970), 227-241. [Annotated and scanned copy]

Crossrefs

Cf. A001200, A056642.

Programs

Mathematica

A001200 = Cases[Import["https://oeis.org/A001200/b001200.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{1} ~Join~ EulerInvTransform[A001200 // Rest] (* Jean-François Alcover, Jan 04 2020, updated Mar 17 2020 *)

Extensions

More terms could be obtained from A056642. - N. J. A. Sloane, Jul 26 2004

a(10)-a(12) from A001200. - Michael Somos, Apr 24 2014

a(12) corrected by Jean-François Alcover, Jan 04 2020