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Number of isomorphism classes of affine Mendelsohn triple systems of order n.
22

%I #43 Apr 17 2023 10:17:19

%S 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 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 0,0,0,0,2,0,1,2,0,0,2,0,5,0,0,2,0

%N Number of isomorphism classes of affine Mendelsohn triple systems of order n.

%C 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 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 The existence spectrum (i.e., n such that a(n)>0) is A003136.

%C Comment from _David Stanovsky_, Mar 19 2022, added by _N. J. A. Sloane_, Mar 20 2022 (Start)

%C This is the sequence a(n) defined in the Donovan et al. paper.

%C The b(n) sequence defined there gives the number of non-affine systems.

%C 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 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 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 Since so many of the initial values of b(n), this does not have its own OEIS entry. (End)

%C 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 David Stanovsky, <a href="/A248107/b248107.txt">Table of n, a(n) for n = 1..1023</a>

%H 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 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 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 (GAP)

%o # For brevity, I do not exploit multiplicativity of a(n) here.

%o a := function(n)

%o local count, gg, g, autg, conj, f, b, x;

%o count := 0;

%o for gg in AllGroups(Size, n, IsAbelian, true) do

%o g := Image(IsomorphismPermGroup(gg), gg);

%o autg := AutomorphismGroup(g);

%o conj := List(ConjugacyClasses(autg), x->Representative(x));

%o for f in conj do

%o b := true;

%o for x in g do

%o if not

%o Image(f, Image(f, x))*Image(f, x^-1)*x = ()

%o then b := false; break;

%o fi;

%o od;

%o if b then count := count + 1; fi;

%o od;

%o od;

%o return count;

%o end;

%Y Cf. A003136, A352550, A352561.

%K nonn,mult

%O 1,7

%A _David Stanovsky_, Oct 01 2014