A090602 -id:A090602 - cp's OEIS frontend

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A090601 Number of n-element groupoids with an identity.

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)

A175882 Number of groupoids of order n with no identity element.

Original entry on oeis.org

1, 0, 12, 19440, 4293918720, 298022460937500000, 10314424627908807366593740800, 256923577502496762167593902921782457650400, 6277101735385253516143083463326608130132905949327651766272, 196627050475552807506414708879663572595247698231546775823411085055396409324160

Offset: 0

Matt Westwood, Dec 05 2010

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..20
ProofWiki, Count of Binary Operations Without Identity

Cf. A002489, A090602.

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

a(n) = n^(n^2) - n^(((n-1)^2)+1) = A002489(n) - A090602(n).

a(0)=1 prepended and terms a(7) and beyond from Andrew Howroyd, Jan 23 2022

Showing 1-2 of 2 results.

cp's OEIS Frontend

A090601 Number of n-element groupoids with an identity.

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