A300221 a(n) is the number of unlabeled, graded rank-3 lattices with n elements.

0, 0, 0, 1, 2, 4, 8, 18, 38, 88, 210, 528, 1396, 3946, 11896, 38644, 135790, 518645, 2160112, 9832013, 48945468, 266458643

Offset: 1

Author

Jukka Kohonen, Mar 01 2018

Keywords

• nonn
• more

Comments

A graded lattice has rank 3 if its maximal chains have length 3.

They can be enumerated with a program such as that by Kohonen (2017).

Also called "two level lattices": apart from top and bottom, they have just coatoms and atoms. (Kleitman and Winston 1980)

Asymptotic upper bound: a(n) < b^(n^(3/2) + o(n^(3/2))), where b is about 1.699408. (Kleitman and Winston 1980)

Examples

			a(4)=1: The only possibility is the chain of length 3 (with 4 elements).
a(6)=4: These are the four lattices.
    o       o      o       o
    |      / \    / \     /|\
    o      o o    o o    o o o
   /|\     | |    |/|     \|/
  o o o    o o    o o      o
   \|/     \ /    \ /      |
    o       o      o       o
		

Links

• D. J. Kleitman and K. J. Winston, The asymptotic number of lattices, Ann. Discrete Math. 6 (1980), 243-249.
• J. Kohonen, Generating modular lattices up to 30 elements, arXiv:1708.03750 [math.CO] preprint (2017).

Crossrefs

Cf. A278691 (unlabeled graded lattices).

Formula

a(n) = Sum_{k=1..n-3} A300260(n-2-k, k).

Extensions

a(22) from Jukka Kohonen, Mar 03 2018