The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 Stanovský and Petr Vojtechovský, 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 27 20:38 EDT 2024. Contains 372882 sequences. (Running on oeis4.)