A369313 Number of interval-closed sets in the boolean lattice of dimension n.

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

Offset: 0

Author

Nadia Lafreniere, Jan 19 2024

Keywords

• nonn
• hard
• more

Comments

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.

Examples

			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}}
		

Links

• Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, and Amanda Welch, Toggling, rowmotion, and homomesy on interval-closed sets, arXiv:2307.08520 [math.CO], 2023.

Crossrefs

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

Programs

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

Extensions

a(6) from Christian Sievers, Jan 27 2024