A090603 -id:A090603 - cp's OEIS frontend

Showing 1-2 of 2 results.

A090602 Number of n-element labeled groupoids with an identity.

0, 1, 4, 243, 1048576, 762939453125, 170581728179578208256, 18562115921017574302453163671207, 1427247692705959881058285969449495136382746624, 106111661199647248543687855752712667991103904330482569981872649

Offset: 0

Christian G. Bower, Dec 05 2003

nonn

Also labeled groupoids with an absorbant (zero) element.

Andrew Howroyd, Table of n, a(n) for n = 0..20
Eric Postpischil Posting to sci.math newsgroup, May 21 1990
Eric Weisstein's World of Mathematics, Groupoid
Wikipedia, Magma (algebra)
Index entries for sequences related to groupoids

Cf. A090601 (isomorphism classes), A090603.

a(n) = n^((n-1)^2+1) \\ Andrew Howroyd, Jan 23 2022

a(n) = n^((n-1)^2+1).

a(n) = A090603(n)*n.

a(0)=0 prepended and a(9) added by Andrew Howroyd, Jan 23 2022

A090601 Number of n-element groupoids with an identity.

Original entry on oeis.org

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

Offset: 1

Christian G. Bower, Dec 05 2003

nonn

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

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

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

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)

Showing 1-2 of 2 results.

cp's OEIS Frontend

A090602 Number of n-element labeled groupoids with an identity.

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