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A300221
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a(n) is the number of unlabeled, graded rank-3 lattices with n elements.
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1
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0, 0, 0, 1, 2, 4, 8, 18, 38, 88, 210, 528, 1396, 3946, 11896, 38644, 135790, 518645, 2160112, 9832013, 48945468, 266458643
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OFFSET
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1,5
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COMMENTS
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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)
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n-3} A300260(n-2-k, k).
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EXAMPLE
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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
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Cf. A278691 (unlabeled graded lattices).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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