cp's OEIS Frontend

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

%I A367316 #10 Jan 26 2024 10:18:30
%S A367316 1,2,8,45,307,2385,20362,186812,1814156,18448851,194918129,2126727740
%N A367316 Number of interval-closed sets in the root poset of type A(n).
%C A367316 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 A367316 Interval-closed sets are also called convex subsets of a poset.
%C A367316 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 A367316 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 A367316 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 A367316 (SageMath)
%o A367316 ICS_count = 0
%o A367316 x = RootSystem(['A',n]).root_poset()
%o A367316 for A in x.antichains_iterator():
%o A367316     I = x.order_ideal(A)
%o A367316     Q = x.subposet(set(I).difference(A))
%o A367316     ICS_count += Q.antichains().cardinality()
%o A367316 ICS_count
%Y A367316 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 A367316 Interval-closed sets for other posets: Cf. A369313, A367109
%K A367316 more,hard,nonn
%O A367316 0,2
%A A367316 _Nadia Lafreniere_, Jan 26 2024