OFFSET
1,5
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)
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).
FORMULA
a(n) = Sum_{k=1..n-3} A300260(n-2-k, k).
EXAMPLE
a(4)=1: The only possibility is the chain of length 3 (with 4 elements).
a(6)=4: These are the four lattices.
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CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jukka Kohonen, Mar 01 2018
EXTENSIONS
a(22) from Jukka Kohonen, Mar 03 2018
STATUS
approved