cp's OEIS Frontend

A248107 Number of isomorphism classes of affine Mendelsohn triple systems of order n.

%I A248107 #43 Apr 17 2023 10:17:19
%S A248107 1,0,1,1,0,0,2,0,2,0,0,1,2,0,0,2,0,0,2,0,2,0,0,0,1,0,3,2,0,0,2,0,0,0,
%T A248107 0,2,2,0,2,0,0,0,2,0,0,0,0,2,5,0,0,2,0,0,0,0,2,0,0,0,2,0,4,3,0,0,2,0,
%U A248107 0,0,0,0,2,0,1,2,0,0,2,0,5,0,0,2,0
%N A248107 Number of isomorphism classes of affine Mendelsohn triple systems of order n.
%C A248107 A Mendelsohn triple system is affine if the associated quasigroup is affine, i.e, given by x*y=(1-f)(x)+f(y) over an abelian group (A,+) with an automorphism f.
%C A248107 For Steiner triple systems, the enumeration is settled by the following observation: a Steiner triple system is affine if and only if A=Z_3^n and f(x)=-x.
%C A248107 The existence spectrum (i.e., n such that a(n)>0) is A003136.
%C A248107 Comment from _David Stanovsky_, Mar 19 2022, added by _N. J. A. Sloane_, Mar 20 2022 (Start)
%C A248107 This is the sequence a(n) defined in the Donovan et al. paper.
%C A248107 The b(n) sequence defined there gives the number of non-affine systems.
%C A248107 The first 728 values of b(n) are now known: they are all zeros, except b(81) = 2, b(243) = 6, b(324) = 2, b(567)=4. We do not know b(729).
%C A248107 The reason is the following: it follows from the Galkin-Fischer-Smith theorem that, for n = m * 3^d, m not divisible by 3, we have b(n) = a(m) * b(3^d).
%C A248107 At the time of writing the paper, we could use known data about commutative Moufang loops to determine b(1) = b(3) = b(9) = b(27) = 0, and b(81) = 2. Later we managed to develop smarter enumeration methods that allowed us to determine b(243)=6: see Jedlička et al. (2007).
%C A248107 Since so many of the initial values of b(n), this does not have its own OEIS entry. (End)
%C A248107 Conjecture: This is the same sequences as A352561.(Note that A352561 has an explicit Dirichlet generating function.) - _N. J. A. Sloane_, Mar 21 2022
%H A248107 David Stanovsky, <a href="/A248107/b248107.txt">Table of n, a(n) for n = 1..1023</a>
%H A248107 Diane M. Donovan, Terry S. Griggs, Thomas A. McCourt, Jakub Opršal, David Stanovský, <a href="http://arxiv.org/abs/1411.5194">Distributive and anti-distributive Mendelsohn triple systems</a>, arXiv:1411.5194 [math.CO], 2014. [Published in Canad. Math. Bull. Vol. 59 (1), 2016 pp. 36-49.] See a(n) on page 9 of arXiv version.
%H A248107 Přemysl Jedlička, David Stanovský, and Petr Vojtěchovský, <a href="https://arxiv.org/abs/1603.00608">Trimedial and distributive quasigroups of order 243</a>, arXiv:1603.00608 [math.GR], 2016.
%H A248107 Přemysl Jedlička, David Stanovský, and Petr Vojtěchovský, <a href="https://doi.org/10.1016/j.disc.2016.08.022">Trimedial and distributive quasigroups of order 243</a>, Discrete Math. 340/3 (2017), 404--415.
%o A248107 (GAP)
%o A248107 # For brevity, I do not exploit multiplicativity of a(n) here.
%o A248107 a := function(n)
%o A248107     local count, gg, g, autg, conj, f, b, x;
%o A248107     count := 0;
%o A248107     for gg in AllGroups(Size, n, IsAbelian, true) do
%o A248107         g := Image(IsomorphismPermGroup(gg), gg);
%o A248107         autg := AutomorphismGroup(g);
%o A248107         conj := List(ConjugacyClasses(autg), x->Representative(x));
%o A248107         for f in conj do
%o A248107             b := true;
%o A248107             for x in g do
%o A248107                 if not
%o A248107                    Image(f, Image(f, x))*Image(f, x^-1)*x = ()
%o A248107                 then b := false; break;
%o A248107                 fi;
%o A248107             od;
%o A248107             if b then count := count + 1; fi;
%o A248107         od;
%o A248107     od;
%o A248107     return count;
%o A248107 end;
%Y A248107 Cf. A003136, A352550, A352561.
%K A248107 nonn,mult
%O A248107 1,7
%A A248107 _David Stanovsky_, Oct 01 2014