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A076017 Number of nonisomorphic systems with n elements with one binary operation satisfying the equation B(AB)=A (semisymmetric quasigroups). 3

%I

%S 1,1,2,3,4,9,41,595,26620,3908953,1867918845

%N Number of nonisomorphic systems with n elements with one binary operation satisfying the equation B(AB)=A (semisymmetric quasigroups).

%C In January of 1968, _Don Knuth_ described the concept of what he called an "abstract grope" to the students in his class for sophomore math majors at Caltech.

%C The students had just learned about abstract groups and he wanted them to get experience doing research with other algebraic axioms; so he challenged them to prove as many interesting things as they could about sets of elements with a binary operator that satisfies the identity x(yx)=y.

%C The name came from the fact that they were groping for results. Such systems were studied in a series of papers by Sade under a more complicated and more dignified yet less memorable name, "semisymmetric quasigroups". The students came up with some good stuff, including the concept of normal subgropes.

%D D. E. Knuth, The Art of Computer Programming, Vol. 4B, in preparation.

%D A. Sade, Quasigroupes demi-symétriques, Ann. Soc. Sci. Bruxelles Sér. I 79 (1965), 133-143.

%H Brendan D. McKay and Ian M. Wanless, <a href="https://doi.org/10.1002/jcd.21814">Enumeration of Latin squares with conjugate symmetry</a>, J. Combin. Des. 30 (2022), 105-130.

%Y Cf. A076016-A076021.

%K nonn,hard,more

%O 1,3

%A Richard C. Schroeppel, Oct 29 2002

%E a(10), a(11) and comments from _Don Knuth_, May 12 2005 - May 14 2005

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