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A369313
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Number of interval-closed sets in the boolean lattice of dimension n.
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2
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OFFSET
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0,1
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COMMENTS
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An interval-closed set of a poset is a subset I such that if x and y are in I with x <= z <= y, then z is in I.
Interval-closed sets are also called convex subsets of a poset.
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LINKS
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EXAMPLE
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The a(0) = 2 through a(2) = 13 interval-closed sets:
{} {} {}
{{}} {{}} {{}}
{{1}} {{1}}
{{}{1}} {{2}}
{{12}}
{{}{1}}
{{}{2}}
{{1}{2}}
{{1}{12}}
{{2}{12}}
{{}{1}{2}}
{{1}{2}{12}}
{{}{1}{2}{12}}
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PROG
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(SageMath)
ICS_count = 0
x = Posets.BooleanLattice(n)
for A in x.antichains_iterator():
I = x.order_ideal(A)
Q = x.subposet(set(I).difference(A))
ICS_count += Q.antichains().cardinality()
ICS_count
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CROSSREFS
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Interval-closed sets are a superset of order ideals. Cf. A000372.
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KEYWORD
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nonn,hard,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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