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A384065 Cardinality of the lattice of order ideals for every order ideal in the lattice of normal subgroups of the dihedral group D_{2*n}.

%I A384065 #11 May 22 2025 22:00:21
%S A384065 3,10,4,11,4,16,4,12,5,16,4,20,4,16,7,13,4,23,4,20,7,16,4,25,5,16,6,
%T A384065 20,4,39,4,14,7,16,7,33,4,16,7,25,4,39,4,20,11,16,4,31,5,23,7,20,4,31,
%U A384065 7,25,7,16,4,69,4,16,11,15,7,39,4,20,7,39,4,48,4,16,11,20,7,39
%N A384065 Cardinality of the lattice of order ideals for every order ideal in the lattice of normal subgroups of the dihedral group D_{2*n}.
%C A384065 An order ideal I is a nonempty subset of a partially ordered set P where y <= x for x in I and y in P implies y is in I. The set of order ideals of a partially ordered set is a lattice (ordered by inclusion) and is distributive.
%C A384065 The lattice of normal subgroups of any group is distributive and also modular.
%C A384065 The number of order ideals in the lattice of normal subgroups of D_{2*n} is a(n)-1 since the empty set is not an order ideal.
%H A384065 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ideal_(order_theory)">Ideal (order theory)</a>
%F A384065 a(n) > A037852(n).
%F A384065 a(p) = 4 for every prime p > 2.
%e A384065 a(1) = 3 since for D_{2} = C_2 the lattice of normal subgroups L = {1, C_2} and the lattice of order ideals of L contains {}, {1}, and L.
%e A384065 a(2) = 10 since for D_{4} = (C_2 x C_2) the lattice of normal subgroups L = {1, C_2, C_2, C_2, (C_2 x C_2)} and the lattice of order ideals of L contains {}, {1}, {1, C_2}, {1, C_2}, {1, C_2}, {1, C_2, C_2}, {1, C_2, C_2}, {1, C_2, C_2}, {1, C_2, C_2, C_2}, and L.
%o A384065 (Sage)
%o A384065 def a(n):
%o A384065     return len(Poset((DihedralGroup(n).normal_subgroups(), lambda H, K: H.is_subgroup(K))).order_ideals_lattice())
%Y A384065 Cf. A037852.
%K A384065 nonn
%O A384065 1,1
%A A384065 _Miles Englezou_, May 18 2025