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A386234 Number of good involutions of all nontrivial core quandles of order n.

%I A386234 #13 Aug 13 2025 18:32:02
%S A386234 1,4,1,3,1,72,2,3,1,31,1,3,1,10856,1,7,1,47,2,3,1
%N A386234 Number of good involutions of all nontrivial core quandles of order n.
%C A386234 A good involution f of a quandle Q is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). We call the pair (Q,f) a symmetric quandle. A symmetric quandle isomorphism is a quandle isomorphism that intertwines good involutions.
%C A386234 A core quandle Core(G) is a group G viewed as a kei (i.e., involutory quandle) under the operation g(h) = g*h^-1*g. Note that Core(G) is nontrivial if and only if exp(G) > 2.
%D A386234 Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, 101-108.
%H A386234 Seiichi Kamada and Kanako Oshiro, <a href="https://doi.org/10.1090/S0002-9947-2010-05131-1">Homology groups of symmetric quandles and cocycle invariants of links and surface-links</a>, Trans. Amer. Math. Soc., 362 (2010), no. 10, 5501-5527.
%H A386234 Lực Ta, <a href="https://arxiv.org/abs/2505.08090">Good involutions of conjugation subquandles</a>, arXiv:2505.08090 [math.GT], 2025. See Table 3.
%H A386234 Lực Ta, <a href="https://github.com/luc-ta/Symmetric-Rack-Classification">Symmetric-Rack-Classification</a>, GitHub, 2025.
%H A386234 <a href="/index/Qua#quandles">Index entries for sequences related to quandles and racks</a>
%F A386234 Let n > 2. Then Ta, Cor. 7.17 implies the following. If n appears in A000040 or A050384, then a(n) = 1. If n appears in A221048, then a(n) = 2. If n > 4 and n appears in A100484, then a(n) = 3.
%e A386234 For n = 4 the only nontrivial core quandle is the dihedral quandle R4 = Core(Z/4Z) of order 4. It is well-known (see Thm. 3.2 of Kamada and Oshiro) that R4 has exactly four good involutions. Hence a(4) = 4.
%e A386234 For n = 6 the only nontrivial core quandles are Core(S3) and R6 = Core(Z/6Z), which have one and two good involutions, respectively. Hence a(6) = 3.
%o A386234 (GAP) See Ta, GitHub link
%Y A386234 Cf. A000001, A386233, A181770, A178432, A386231, A386232.
%K A386234 nonn,hard,more
%O A386234 3,2
%A A386234 _Luc Ta_, Jul 21 2025