A038017 Number of n-element commutative groupoids with an identity ("pointed" groupoids).

1, 2, 15, 720, 409600, 3920030472, 775775333825891, 3837862827737186253664, 558740081065710564284870598075, 2755731923933734753149997221152548428020, 520996314135332606285488148844494695722050333912483

Offset: 1

Author

Christian G. Bower, May 15 1998; revised Dec 05 2003

Keywords

• nonn

Comments

Also number of commutative partial groupoids with n-1 elements or commutative groupoids with an absorbant (zero) element with n elements.

Links

• Eric Postpischil Posting to sci.math newsgroup, May 21 1990
• Eric Weisstein's World of Mathematics, Groupoid.
• Index entries for sequences related to groupoids

Crossrefs

Cf. A001329, A030257.

Formula

a(n+1) = 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, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (1 + sum {d|i} (d*s_d))^((i*s_i^2+s_i)/2) or {i=j, even} (1 + sum {d|i} (d*s_d))^(i*s_i^2/2) * (1 + sum {d|i/2} (d*s_d))^s_i or {i != j} (1 + sum {d|lcm(i, j)} (d*s_d))^(2*gcd(i, j)*s_i*s_j)

a(n) asymptotic to (n^binomial(n, 2)+1)/n! = A090599(n)/A000142(n) = A076113(n)/A000142(n-1)