%I #26 Nov 24 2023 07:23:59

%S 1,1,5,13,19,5,41,73,116,19,109,65,155,41,95,669,271,116,341,247,205,

%T 109,505,365,1084,155,1574,533,811,95,929,4193,545,271,779,1508,1331,

%U 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

%N The number of medial quasigroups of order n, up to isomorphism.

%C 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.

%H David Stanovsky, <a href="/A226193/b226193.txt">Table of n, a(n) for n = 1..63</a>

%H David Stanovský and Petr Vojtechovský, <a href="http://arxiv.org/abs/1511.03534">Central and medial quasigroups of small order</a>, arxiv preprint arXiv:1511.03534 [math.GR], 2015.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Medial">Medial</a>

%p a:=proc(n)

%p if n = 1 then

%p return 1;

%p else

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

%p end if;

%p end proc;

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

%o LoadPackage("loops");

%o MQ := function( n, k )

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

%o G := SmallGroup( n, k );

%o G := IntoLoop( G );

%o ct := CayleyTable( G );

%o elms := Elements( G );

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

%o A := AutomorphismGroup( G );

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

%o count := 0;

%o for f in f_reps do

%o Cf := Centralizer( A, f );

%o O := OrbitsDomain( Cf, A );

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

%o for g in g_reps do

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

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

%o unused := [1..n];

%o while not IsEmpty( unused ) do

%o c := unused[1];

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

%o if Size(W) = Length(unused) then

%o unused := [];

%o else

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

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

%o unused := Difference( unused, Wc );

%o fi;

%o od;

%o od;

%o od;

%o return count;

%o end;

%o # _David Stanovsky_, Nov 12 2015

%K nonn,hard,mult

%O 1,3

%A _W. Edwin Clark_, May 30 2013

%E a(9)-a(63) from _David Stanovsky_, Nov 12 2015