A281574 Number of geometric lattices on n nodes.

1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 3, 5, 3, 4, 5, 6, 6, 8, 9, 16, 16, 21, 29, 45, 50, 95, 136, 220, 392, 680, 1270, 2530, 4991

Offset: 1

Author

Jukka Kohonen, Jan 24 2017

Keywords

• nonn
• more
• hard

Comments

A finite lattice is geometric if it is semimodular and atomistic. Atomistic (or atomic in Stanley's terminology) means that every element is a join of some atoms; or equivalently, that every join-irreducible element is an atom.

a(n) is the number of simple matroids with n flats, up to isomorphism. - Harry Richman, Jul 27 2022

Examples

			From _Peter Luschny_, Jan 24 2017: (Start)
The only two geometric lattices on 8 nodes:
            7
          / | \
         /  |  \            _ _ 7_ _
         3  5  6           / / /\ \ \
         |\/ \/|          / / /  \ \ \
         |/\ /\|         1 2 3    4 5 6
         1  2  4          \ \ \  / / /
          \ | /            \_\_\/_/_/
           \|/                  0
            0
(End)
		

Links

• J. Kohonen, Generating modular lattices up to 30 elements, arXiv:1708.03750 [math.CO], 2017-2018.
• M. Malandro, The unlabeled lattices on <=15 nodes, (listing of lattices; geometric lattices are a subset of these).
• Wikipedia, Geometric lattice

Crossrefs

Cf. A229202 (semimodular lattices).

Extensions

a(16)-a(34) from Kohonen (2017), by Jukka Kohonen, Aug 15 2017

a(35)-a(37) by Jukka Kohonen, Jul 07 2020