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

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, 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, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 5, 0, 0, 2, 0

Offset: 1

Author

David Stanovsky, Oct 01 2014

Keywords

• nonn
• mult

Comments

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.

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.

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

Comment from David Stanovsky, Mar 19 2022, added by N. J. A. Sloane, Mar 20 2022 (Start)

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

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

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).

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).

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).

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

Conjecture: This is the same sequences as A352561.(Note that A352561 has an explicit Dirichlet generating function.) - N. J. A. Sloane, Mar 21 2022

Links

• David Stanovsky, Table of n, a(n) for n = 1..1023
• Diane M. Donovan, Terry S. Griggs, Thomas A. McCourt, Jakub Opršal, David Stanovský, Distributive and anti-distributive Mendelsohn triple systems, 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.
• Přemysl Jedlička, David Stanovský, and Petr Vojtěchovský, Trimedial and distributive quasigroups of order 243, arXiv:1603.00608 [math.GR], 2016.
• Přemysl Jedlička, David Stanovský, and Petr Vojtěchovský, Trimedial and distributive quasigroups of order 243, Discrete Math. 340/3 (2017), 404--415.

Crossrefs

Cf. A003136, A352550, A352561.

Programs

GAP

# For brevity, I do not exploit multiplicativity of a(n) here.
a := function(n)
    local count, gg, g, autg, conj, f, b, x;
    count := 0;
    for gg in AllGroups(Size, n, IsAbelian, true) do
        g := Image(IsomorphismPermGroup(gg), gg);
        autg := AutomorphismGroup(g);
        conj := List(ConjugacyClasses(autg), x->Representative(x));
        for f in conj do
            b := true;
            for x in g do
                if not
                   Image(f, Image(f, x))*Image(f, x^-1)*x = ()
                then b := false; break;
                fi;
            od;
            if b then count := count + 1; fi;
        od;
    od;
    return count;
end;