A367316 Number of interval-closed sets in the root poset of type A(n).

1, 2, 8, 45, 307, 2385, 20362, 186812, 1814156, 18448851, 194918129, 2126727740

Offset: 0

Author

Nadia Lafreniere, Jan 26 2024

Keywords

• more
• hard
• nonn

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.

The root poset of a root system is the partial order on positive roots where a <= b if b-a is a nonnegative sum of simple roots.

Examples

			For n = 0, the poset is empty, so there is only one subset.For n = 1, the poset has only one element, and both subsets are interval-closed.For n = 2, the poset has three elements, and rank 1. Every subset of a poset of rank at most 1 is interval-closed, and therefore there are a(2) = 8 interval-closed sets.For n = 3, the poset has six elements, and only 45 of the 64 subsets are interval-closed.
		

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. Order ideals of the type A root poset are counted by the Catalan numbers. Cf. A000108

Interval-closed sets for other posets: Cf. A369313, A367109

Programs

SageMath

ICS_count = 0
x = RootSystem(['A',n]).root_poset()
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