A079190 Number of isomorphism classes of anti-commutative closed binary operations (groupoids) on a set of order n.

1, 6, 996, 31857648, 266666713602640, 929809173755713574913480, 2002123402266181527640478418179038176, 3702236248557739850415303240942330019881771301360640, 7805296829528400289943264314587254996361382902046539931447903763389056

Offset: 1

Author

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

Keywords

• nonn

Comments

A079187(n)+A079190(n)=A001329(n).

Each a(n) is equal to the sum of the elements in row n of A079191.

Links

• C. van den Bosch, Closed binary operations on small sets
• Index entries for sequences related to groupoids

Crossrefs

Cf. A079187, A079189, A079191.

Formula

a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = Product_{i>=1, j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (Sum_{d|i} (d*s_d))^(s_i*(i*s_i+1)/2) * (-1 + Sum_{d|i} (d*s_d))^(s_i*(i*s_i-1)/2) or {i=j, even} (Sum_{d|i and i/d is odd} (d*s_d))^s_i * (Sum_{d|i} (d*s_d))^(i*s_i^2/2) * (-1 + Sum_{d|i} (d*s_d))^(s_i*(i*s_i-2)/2) or {i < j} (Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j) or {i > j} (-1 + Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j). [Corrected by Sean A. Irvine, Aug 03 2025]

a(n) is asymptotic to (n^binomial(n+1, 2) * (n-1)^binomial(n, 2))/n! = A079189(n)/A000142(n)

Extensions

Edited, corrected and extended with formula by Christian G. Bower, Dec 12 2003

a(9) from Sean A. Irvine, Aug 03 2025