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Number of interval-closed sets in the boolean lattice of dimension n.
2

%I #20 Jan 28 2024 01:45:39

%S 2,4,13,101,3938,3257610,676675164063

%N Number of interval-closed sets in the boolean lattice of dimension n.

%C 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.

%C Interval-closed sets are also called convex subsets of a poset.

%H Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, and Amanda Welch, <a href="http://arxiv.org/abs/2307.08520">Toggling, rowmotion, and homomesy on interval-closed sets</a>, arXiv:2307.08520 [math.CO], 2023.

%e The a(0) = 2 through a(2) = 13 interval-closed sets:

%e {} {} {}

%e {{}} {{}} {{}}

%e {{1}} {{1}}

%e {{}{1}} {{2}}

%e {{12}}

%e {{}{1}}

%e {{}{2}}

%e {{1}{2}}

%e {{1}{12}}

%e {{2}{12}}

%e {{}{1}{2}}

%e {{1}{2}{12}}

%e {{}{1}{2}{12}}

%o (SageMath)

%o ICS_count = 0

%o x = Posets.BooleanLattice(n)

%o for A in x.antichains_iterator():

%o I = x.order_ideal(A)

%o Q = x.subposet(set(I).difference(A))

%o ICS_count += Q.antichains().cardinality()

%o ICS_count

%Y Interval-closed sets are a superset of order ideals. Cf. A000372.

%K nonn,hard,more

%O 0,1

%A _Nadia Lafreniere_, Jan 19 2024

%E a(6) from _Christian Sievers_, Jan 27 2024