A090601 Number of n-element groupoids with an identity.

1, 2, 45, 43968, 6358196250, 236919104155855296, 3682959509036574988532481464, 35398008251644050232134479709365068115968, 292415292106611727928759157427747328169866020125762652311

Offset: 1

Author

Christian G. Bower, Dec 05 2003

Keywords

• nonn

Comments

Also partial groupoids with n-1 elements or 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

Programs

Sage

R. = InfinitePolynomialRing(QQ)
@cached_function
def Z(n):
    if n==0:
        return R.one()
    return sum(a[k]*Z(n-k) for k in (1..n))/n
def magmas_identity(n):
    P = Z(n-1)
    q = 0
    c = P.coefficients()
    count = 0
    for m in P.monomials():
        r = 1
        T = m.variables()
        S = list(T)
        for u in T:
            i = R.varname_key(str(u))[1]
            j = m.degree(u)
            D = 1
            for d in divisors(i):
                D += d*m.degrees()[-d-1]
            r *= D^(i*j^2)
            S.remove(u)
            for v in S:
                k = R.varname_key(str(v))[1]
                l = m.degree(v)
                D = 1
                for d in divisors(lcm(i,k)):
                    try:
                        D += d*m.degrees()[-d-1]
                    except:
                        break
                r *= D^(gcd(i,k)*j*l*2)
        q += c[count]*r
        count += 1
    return q
# Philip Turecek, Oct 10 2023

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} ( (1 + sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j))

a(n) asymptotic to n^((n-1)^2+1)/n! = A090602(n)/A000142(n) = A090603(n)/A000142(n-1)