A367316 - OEIS

%I #10 Jan 26 2024 10:18:30

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

%N Number of interval-closed sets in the root poset of type A(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.

%C 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.

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

%e 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.

%o (SageMath)

%o ICS_count = 0

%o x = RootSystem(['A',n]).root_poset()

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

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

%K more,hard,nonn

%O 0,2

%A _Nadia Lafreniere_, Jan 26 2024