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A248107 Number of isomorphism classes of affine Mendelsohn triple systems of order n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

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.

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.

PROG

(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;

CROSSREFS

Cf. A003136.

Sequence in context: A035147 A101673 A091395 * A035220 A227618 A221645

Adjacent sequences:  A248104 A248105 A248106 * A248108 A248109 A248110

KEYWORD

nonn,mult

AUTHOR

David Stanovsky, Oct 01 2014

STATUS

approved

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Last modified April 20 11:15 EDT 2019. Contains 322309 sequences. (Running on oeis4.)