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A367316
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Number of interval-closed sets in the root poset of type A(n).
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0
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1, 2, 8, 45, 307, 2385, 20362, 186812, 1814156, 18448851, 194918129, 2126727740
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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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
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CROSSREFS
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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
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KEYWORD
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more,hard,nonn
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AUTHOR
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STATUS
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approved
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