cp's OEIS Frontend

A248908 The number of isomorphism classes of Latin keis (involutory right distributive quasigroups) of order n.

%I A248908 #21 Jun 03 2018 03:44:44
%S A248908 1,0,1,0,1,0,1,0,2,0,1,0,1,0,2,0,1,0,1,0,2,0,1,0,2,0,7,0,1,0,1,0,2,0,
%T A248908 1,0,1,0,2,0,1,0,1,0,5,0,1
%N A248908 The number of isomorphism classes of Latin keis (involutory right distributive quasigroups) of order n.
%C A248908 A quandle (Q,*) is a kei or involutory quandle if for all x,y in Q we have (x*y)*y = x, that is, all right translations R_a: x-> x*a, are involutions. A quandle (Q,*) is a quasigroup if also the mappings L_a: x->a*x are bijections.
%C A248908 Masahico Saito noticed that a(n) = 0 if n is even. Here is a simple proof: Suppose that Q is a Latin kei of order n and that n is even. Let R_a be the permutation of Q given by R_a(x) = x*a. Since R_a is an involution it is a product of t transpositions. Let f be the number of fixed points of R_a. Then n = 2*t + f. Since R_a(a) = a and n is even, there must be a fixed point x different from a. Hence x*a = x and x*x = x. So L_x is not a bijection. This shows that Q is not Latin, so the result is proved.
%C A248908 a(n) > 0 if n is odd: Consider the Latin kei defined on Z/(n) by the rule x*y = -x + 2y.
%C A248908 Leandro Vendramin (see link below) has found all connected quandles of order n for n at most 47. (There are 790 of them, not counting the one of order 1.) A Latin quandle is connected. So this sequence was found by just going through Vendramin's list and counting the quandles which are Latin keis.
%H A248908 Scott Carter, <a href="http://arxiv.org/abs/1002.4429"> A Survey of Quandle Ideas</a>, arXiv:1002.4429 [math.GT], 2010.
%H A248908 Leandro Vendramin and Matías Graña,<a href="https://code.google.com/p/rig/"> Rig, a GAP package for racks and quandles.</a>
%H A248908 Leandro Vendramin,<a href="http://arxiv.org/abs/1105.5341"> On the classification of quandles of low order</a>, arXiv:1105.5341 [math.GT], 2011-2012.
%H A248908 S. K. Stein, <a href="http://dx.doi.org/10.1090/S0002-9947-1957-0094404-6"> On the Foundations of Quasigroups</a>, Transactions of American Mathematical Society, 85 (1957), 228-256.
%Y A248908 Dropping all zeros gives A254434.
%K A248908 nonn,hard,more
%O A248908 1,9
%A A248908 _W. Edwin Clark_, Mar 06 2015