A001329 Number of nonisomorphic groupoids with n elements.

1, 1, 10, 3330, 178981952, 2483527537094825, 14325590003318891522275680, 50976900301814584087291487087214170039, 155682086691137947272042502251643461917498835481022016

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

The number of isomorphism classes of closed binary operations on a set of order n.

The term "magma" is also used as an alternative for "groupoid" since the latter has a different meaning in e.g. category theory. - Joel Brennan, Jan 20 2022

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Philip Tureček, Table of n, a(n) for n = 0..27
• J. Berman and S. Burris, A computer study of 3-element groupoids Lect. Notes Pure Appl. Math. 180 (1994) 379-429 MR1404949
• M. A. Harrison, The number of isomorphism types of finite algebras, Proc. Amer. Math. Soc., 17 (1966), 731-737.
• Eric Postpischil, Posting to sci.math newsgroup, May 21 1990
• Marko Riedel, Counting non-isomorphic binary relations, Mathematics Stack Exchange question.
• N. J. A. Sloane, Overview of A001329, A001423-A001428, A258719, A258720.
• T. Tamura, Some contributions of computation to semigroups and groupoids, pp. 229-261 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970. (Annotated and scanned copy)
• Philip Turecek, Maple program
• Philip Tureček, Counting Finite Magmas, arXiv:2305.00269 [math.CO], 2023.
• Eric Weisstein's World of Mathematics, Groupoid
• Wikipedia, Magma (algebra)
• Index entries for sequences related to groupoids

Crossrefs

Cf. A001424, A001425, A002489, A006448, A029850, A030245-A030265, A030271, A038015-A038023.

Programs

Maple

with(numtheory);
with(group):
with(combinat):
pet_cycleind_symm :=
proc(n)
        local p, s;
        option remember;
        if n=0 then return 1; fi;
        expand(1/n*add(a[l]*pet_cycleind_symm(n-l), l=1..n));
end;
pet_flatten_term :=
proc(varp)
        local terml, d, cf, v;
        terml := [];
        cf := varp;
        for v in indets(varp) do
            d := degree(varp, v);
            terml := [op(terml), seq(v, k=1..d)];
            cf := cf/v^d;
        od;
        [cf, terml];
end;
bs_binop :=
proc(n)
        option remember;
        local dsjc, flat, p, q, len,
              cyc, cyc1, cyc2, l1, l2, res;
        if n=0 then return 1; fi;
        if n=1 then return 1; fi;
        res := 0;
        for dsjc in pet_cycleind_symm(n) do
            flat := pet_flatten_term(dsjc);
            p := 1;
            for cyc1 in flat[2] do
                l1 := op(1, cyc1);
                for cyc2 in flat[2] do
                    l2 := op(1, cyc2);
                    len := lcm(l1,l2); q := 0;
                    for cyc in flat[2] do
                        if len mod op(1, cyc) = 0 then
                           q := q  + op(1, cyc);
                        fi;
                    od;
                    p := p * q^(l1*l2/len);
                od;
            od;
            res := res + p*flat[1];
        od;
        res;
end;

Mathematica

magmas[n_] := (
    rul1 = {{a[i_],j_},{a[k_],l_}} :> sum[i,k]^(j*l*GCD[i,k]*(2-Boole[i==k]));
    rul2 = {a[r_],s_} :> If[Mod[lcm,r]==0,r*s,0];
    trans[mo_] := (
        listc = FactorList@mo;
        list = listc[[2;;]];
        sum[i_,k_] := (
            lcm = LCM[i,k];
            Plus@@(list/.rul2)
        );
        pairs = Select[Tuples[list,2],OrderedQ];
        listc[[1,1]]^listc[[1,2]]*Times@@(pairs/.rul1)
    );
    trans/@CycleIndexPolynomial[SymmetricGroup@n,Array[a,n]]
);
(* Philip Turecek, May 25 2022 *)

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(n):
    P = Z(n)
    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 = 0
            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 = 0
                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, Nov 20 2022

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, j>=1} ( (Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j)). - Christian G. Bower, May 08 1998, Dec 03 2003

a(n) is asymptotic to n^(n^2)/n! = A002489(n)/A000142(n) ~ (e*n^(n-1))^n / sqrt(2*Pi*n). - Christian G. Bower, Dec 03 2003

a(n) = A079173(n) + A027851(n) = A079177(n) + A079180(n).

a(n) = A079183(n) + A001425(n) = A079187(n) + A079190(n).

a(n) = A079193(n) + A079196(n) + A079199(n) + A001426(n).

Extensions

More terms from Christian G. Bower, May 08 1998