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

1,2

Comments

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

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

Links

Table of n, a(n) for n=1..9.

C. van den Bosch, Closed binary operations on small sets

Index entries for sequences related to groupoids

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)

Crossrefs

Cf. A079187, A079189, A079191.

Sequence in context: A375020 A024085 A080474 * A203303 A159865 A370680

Adjacent sequences: A079187 A079188 A079189 * A079191 A079192 A079193

Keyword

nonn

Author

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

Extensions

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

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

Status

approved