OFFSET
0,1
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.
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.
EXAMPLE
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}}
PROG
(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
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Nadia Lafreniere, Jan 19 2024
EXTENSIONS
a(6) from Christian Sievers, Jan 27 2024
STATUS
approved