A079243 Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.

0, 0, 2, 2, 3, 2, 4, 2, 4

Offset: 0

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

• nonn
• more

Comments

The only closed binary operations that are both commutative and anti-commutative are those on sets of size <= 1. The significance of non-commutative (and non-anti-associative) in the name is that it excludes this possibility. Otherwise, the first two terms would be 1. - Andrew Howroyd, Jan 26 2022

Examples

			From _Andrew Howroyd_, Jan 26 2022: (Start)
The a(6) = 4 operations are the two shown below and their converses.
    | 1 2 3 4 5 6         | 1 2 3 4 5 6
  --+------------       --+------------
  1 | 1 2 3 4 5 6       1 | 1 2 3 1 2 3
  2 | 1 2 3 4 5 6       2 | 1 2 3 1 2 3
  3 | 1 2 3 4 5 6       3 | 1 2 3 1 2 3
  4 | 1 2 3 4 5 6       4 | 4 5 6 4 5 6
  5 | 1 2 3 4 5 6       5 | 4 5 6 4 5 6
  6 | 1 2 3 4 5 6       6 | 4 5 6 4 5 6
(End)
		

Links

• Index entries for sequences related to semigroups

Crossrefs

Row sums of A079208.

Cf. A079242 (labeled case), A079231, A079233, A079235, A079237, A079196, A079241, A079245, A063524.

Formula

A079231(n) + A079233(n) + A079235(n) + A079237(n) + A079196(n) + A079241(n) + a(n) + A079245(n) + A063524(n) = A002489(n).

Conjecture: a(n) = A000005(n) for n > 1. - Andrew Howroyd, Jan 26 2022

Extensions

a(0)=0 prepended and a(5)-a(8) from Andrew Howroyd, Jan 26 2022