A079187 -id:A079187 - cp's OEIS frontend

A001329 Number of nonisomorphic groupoids with n elements.

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

Offset: 0

N. J. A. Sloane

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

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).

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

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

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;

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 *)

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

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).

More terms from Christian G. Bower, May 08 1998

A079190 Number of isomorphism classes of anti-commutative closed binary operations (groupoids) on a set of order n.

Original entry on oeis.org

1, 6, 996, 31857648, 266666713602640, 929809173755713574913480, 2002123402266181527640478418179038176, 3702236248557739850415303240942330019881771301360640, 7805296829528400289943264314587254996361382902046539931447903763389056

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

nonn

A079187(n)+A079190(n)=A001329(n).

Each a(n) is equal to the sum of the elements in row n of A079191.

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A079187, A079189, A079191.

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>=1, j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (Sum_{d|i} (d*s_d))^(s_i*(i*s_i+1)/2) * (-1 + Sum_{d|i} (d*s_d))^(s_i*(i*s_i-1)/2) or {i=j, even} (Sum_{d|i and i/d is odd} (d*s_d))^s_i * (Sum_{d|i} (d*s_d))^(i*s_i^2/2) * (-1 + Sum_{d|i} (d*s_d))^(s_i*(i*s_i-2)/2) or {i < j} (Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j) or {i > j} (-1 + Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j). [Corrected by Sean A. Irvine, Aug 03 2025]

a(n) is asymptotic to (n^binomial(n+1, 2) * (n-1)^binomial(n, 2))/n! = A079189(n)/A000142(n)

Edited, corrected and extended with formula by Christian G. Bower, Dec 12 2003

a(9) from Sean A. Irvine, Aug 03 2025

A079186 Number of non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 8, 13851, 3530555392, 266023223876953125, 9644962193498535546171949056, 246832875573638552740275218239438131202951, 6127827569844832702316847785612357470342156990019367075840, 193794664362053647720926884692597177807303542565053791345764052714030485961865

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..20
C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A002489, A079187 (non-isomorphic), A079188, A079189, A079210.

a(n) = n^(n^2) - (n^n)*((n^2-n)^((n^2-n)/2)) \\ Andrew Howroyd, Jan 23 2022

a(n) = n^(n^2) - (n^n)*((n^2-n)^((n^2-n)/2)).
a(n) = A002489(n) - A079189(n).
a(n) = Sum_{k>=1} A079178(n,k)*A079210(n,k).

a(0)=0 prepended and terms a(5) and beyond from Andrew Howroyd, Jan 23 2022

A079188 Number of isomorphism classes of non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 4, 1, 4, 44, 2285, 0, 0, 0, 24, 64, 212, 35240, 147088764

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
A079176(n) is equal to the sum of the products of each element in row n of this sequence and the corresponding element of A079210.
The sum of each row n of this sequence is given by A079177(n).

			First four rows:
  0;
  0, 4;
  1, 4, 44, 2285;
  0, 0, 0, 24, 64, 212, 35240, 147088764.

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A027423, A079171, A079186, A079187, A079191.

a(n) = A079171(n) - A079191(n).

A118542 Number of nonisomorphic groupoids with <= n elements.

Original entry on oeis.org

1, 2, 12, 3342, 178985294, 2483527716080119, 14325590005802419238355799, 50976900301828909677297289506452525838, 155682086691137998248942804080553139214788341933547854

Offset: 0

Jonathan Vos Post, May 06 2006

nonn

The number of isomorphism classes of closed binary operations on sets of order <= n. See formulas by Christian G. Bower in A001329 Number of nonisomorphic groupoids with n elements.

			a(5) = 1 + 1 + 10 + 3330 + 178981952 + 2483527537094825 = 2483527716080119 is prime.

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

a(n) = SUM[i=0..n] A001329(i). a(n) = SUM[i=0..n] (A079173(i)+A027851(i)). a(n) = SUM[i=0..n] (A079177(i)+A079180(i)). a(n) = SUM[i=0..n] (A079183(i)+A001425(i)). a(n) = SUM[i=0..n] (A079187(i)+A079190(i)). a(n) = SUM[i=0..n] (A079193(i)+A079196(i)+A079199(i)+A001426(i)).

cp's OEIS Frontend

A001329 Number of nonisomorphic groupoids with n elements.

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A079190 Number of isomorphism classes of anti-commutative closed binary operations (groupoids) on a set of order n.

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A079186 Number of non-anti-commutative closed binary operations on a set of order n.

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A079188 Number of isomorphism classes of non-anti-commutative closed binary operations on a set of order n, listed by class size.

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A118542 Number of nonisomorphic groupoids with <= n elements.

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