A076021 -id:A076021 - cp's OEIS frontend

A076016 Number of systems with n elements having one binary operation satisfying the equation B(AB)=A (semisymmetric quasigroups).

1, 2, 3, 18, 120, 2880, 140256, 20782080, 9569532672, 14175610675200, 74559788174868480

Offset: 1

Richard C. Schroeppel, Oct 29 2002

Or, Latin squares for which the set of (row,column,symbol) triples is invariant under cyclic permutation of the elements within each triple.

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076017-A076021, A350019, A350020.

a(10)-a(11) from Ian Wanless, Dec 08 2021

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

Original entry on oeis.org

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

Offset: 1

Richard C. Schroeppel, Oct 29 2002

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.
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.
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. E. Knuth, The Art of Computer Programming, Vol. 4B, in preparation.
A. Sade, Quasigroupes demi-symétriques, Ann. Soc. Sci. Bruxelles Sér. I 79 (1965), 133-143.

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076016-A076021.

a(10), a(11) and comments from Don Knuth, May 12 2005 - May 14 2005

A076019 Number of nonisomorphic commutative systems with n elements with one binary operation satisfying the equation B(AB)=A (totally-symmetric quasigroups).

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 13, 12, 139, 65, 25894, 24316, 92798256, 122859802, 4366600209354

Offset: 1

Richard C. Schroeppel, Oct 29 2002

Hy Ginsberg, Totally Symmetric Quasigroups of Order 16, arXiv preprint (2022). arXiv:2211.13204 [math.CO]
Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076016-A076021.

a(10)-a(15) from Ian Wanless, Dec 08 2021

a(16) from Ginsberg link via Charles R Greathouse IV, Dec 02 2022

A350022 Number of idempotent semisymmetric Latin squares of order n.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 480, 0, 2274048, 757555200, 4693077997977600

Offset: 1

Ian Wanless, Dec 08 2021

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076021, A350023, A350024.

A350024 Number of diagonal semisymmetric quasigroups of order n.

Original entry on oeis.org

1, 0, 2, 1, 1, 0, 7, 2, 112, 2369, 347299, 237570420

Offset: 1

Ian Wanless, Dec 08 2021

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076021, A350022, A350023.

A076020 Number of systems with n elements with one binary operation satisfying the equation B(AB)=A (semisymmetric quasigroups). This gives numbers of systems which are isomorphic to their transpose but are noncommutative.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 24, 178, 1328

Offset: 1

Richard C. Schroeppel, Oct 29 2002

Cf. A076016-A076021.

A076018 Number of systems with n elements with one binary operation satisfying the equation B(AB)=A (semisymmetric quasigroups). Isomorphic systems and systems differing by a transposition have been omitted.

Original entry on oeis.org

1, 1, 2, 3, 4, 9, 34, 393, 13980

Offset: 1

Richard C. Schroeppel, Oct 29 2002

Cf. A076016-A076021.

cp's OEIS Frontend

A076016 Number of systems with n elements having one binary operation satisfying the equation B(AB)=A (semisymmetric quasigroups).

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

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A076019 Number of nonisomorphic commutative systems with n elements with one binary operation satisfying the equation B(AB)=A (totally-symmetric quasigroups).

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A350022 Number of idempotent semisymmetric Latin squares of order n.

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A350024 Number of diagonal semisymmetric quasigroups of order n.

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A076020 Number of systems with n elements with one binary operation satisfying the equation B(AB)=A (semisymmetric quasigroups). This gives numbers of systems which are isomorphic to their transpose but are noncommutative.

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A076018 Number of systems with n elements with one binary operation satisfying the equation B(AB)=A (semisymmetric quasigroups). Isomorphic systems and systems differing by a transposition have been omitted.

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