A350875 a(n) is the number of nonisomorphic left-right-alternative magmas with n elements. That is, a(n) is the number of nonisomorphic magmas with n elements which satisfy the identities (xx)y = x(xy) and x(yy) = (xy)y for all x and y.

1, 1, 5, 43, 2027, 1005972

Offset: 0

Author

Joel Brennan, Jan 20 2022

Keywords

• nonn
• hard
• more

Comments

Compare with A350876, whose terms are smaller (for n > 2) - this means that the left and right alternative identities (xx)y = x(xy) and x(yy) = (xy)y do not imply the flexible identity (xy)x = x(yx) for magmas. This is in contrast to the situation for non-associative rings, where left-right-alternativity implies flexibility (due to the additional additive structure).

a(n) = A350874(n) for n <= 2, i.e., a magma with (zero, one or) two elements which is left (resp., right) alternative is also right (resp., left) alternative.

Examples

			There are 10 nonisomorphic magmas with 2 elements, 5 of which are left-right-alternative, so a(2) = 5.
Similarly there are 3330 nonisomorphic magmas with 3 elements, 43 of which are left-right-alternative, so a(3) = 43.
		

Crossrefs

Cf. A001329 (magmas), A350874 (left/right-alternative magmas), A350876, A350873.

Extensions

a(5) from Andrew Howroyd, Jan 29 2022