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
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
Programs
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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
Comments