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A226193 The number of medial quasigroups of order n, up to isomorphism. 2
1, 1, 5, 13, 19, 5, 41, 73, 116, 19, 109, 65, 155, 41, 95, 669, 271, 116, 341, 247, 205, 109, 505, 365, 1084, 155, 1574, 533, 811, 95, 929, 4193, 545, 271, 779, 1508, 1331, 341, 775, 1387, 1639, 205, 1805, 1417, 2204, 505, 2161, 3345, 4388, 1084, 1355, 2015, 2755, 1574, 2071, 2993, 1705, 811, 3421, 1235, 3659, 929, 4756 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

See the Wikipedia link for "Medial" for definitions. This article also contains the Bruck-Toyoda theorem which characterizes medial quasigroups in terms of abelian groups.

LINKS

David Stanovsky, Table of n, a(n) for n = 1..63

David Stanovsk√Ĺ, Petr Vojtechovsk√Ĺ, Central and medial quasigroups of small order, arxiv preprint arXiv:1511.03534 [math.GR], 2015.

Wikipedia, Medial

MAPLE

a:=proc(n)

if n = 1 then

     return 1;

  else

return MAGMA:-Enumerate(n, 'medial', 'quasigroup');

end if;

end proc;

PROG

# GAP, gives the number of medial quasigroups over SmallGroup(n, k)

LoadPackage("loops");

MQ := function( n, k )

    local G, ct, elms, inv, A, f_reps, count, f, Cf, O, g_reps, g, Cfg, W, unused, c, Wc;

    G := SmallGroup( n, k );

    G := IntoLoop( G );

    ct := CayleyTable( G );

    elms := Elements( G );

    inv := List( List( [1..n], i -> elms[i]^(-1) ), x -> x![1] );

    A := AutomorphismGroup( G );

    f_reps := List( ConjugacyClasses( A ), Representative );

    count := 0;

    for f in f_reps do

        Cf := Centralizer( A, f );

        O := OrbitsDomain( Cf, A );

        g_reps := List( O, x -> x[1] );

        for g in g_reps do

            Cfg := Intersection( Cf, Centralizer( A, g ) );

            W := Set( [1..n], w -> ct[w][ inv[ ct[w^f][w^g] ] ] );

            unused := [1..n];

            while not IsEmpty( unused ) do

                c := unused[1];

                if f*g=g*f then count := count + 1; fi;

                if Size(W) = Length(unused) then

                    unused := [];

                else

                    Wc := Set( W, w -> ct[w][c] );

                    Wc := Union( Orbits( Cfg, Wc ) );

                    unused := Difference( unused, Wc );

                fi;

            od;

        od;

    od;

    return count;

end;

# David Stanovsky, Nov 12 2015

CROSSREFS

Sequence in context: A190432 A197563 A022138 * A028274 A272723 A245177

Adjacent sequences:  A226190 A226191 A226192 * A226194 A226195 A226196

KEYWORD

nonn,hard,mult

AUTHOR

W. Edwin Clark, May 30 2013

EXTENSIONS

a(9)-a(63) from David Stanovsky, Nov 12 2015

STATUS

approved

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Last modified December 9 16:42 EST 2019. Contains 329879 sequences. (Running on oeis4.)