%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