A001424 -id:A001424 - 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

A258720 Number of non-self-dual groupoids which are equal to their duals.

Original entry on oeis.org

0, 3, 1596, 89460896, 1241763541901150, 7162795001623210170643008, 25488450150907291894372809845206177481, 77841043345568973636021232493841647618443964915982324, 270925719901279918478856582434909122129159229348142178651137261056627814

Offset: 1

N. J. A. Sloane, Jun 18 2015

nonn

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)

Cf. A001329, A001424, A029850, A258719.

a(n) = (A001329(n) - A029850(n))/2. - Andrew Howroyd, May 06 2023

Terms a(5) and beyond from Andrew Howroyd, May 06 2023

A029850 Number of self-converse groupoids.

Original entry on oeis.org

1, 1, 4, 138, 60160, 453292525, 72471180989664, 298545867396801815077, 37263960166680610905649057368, 161614516495439236943507628117344255307, 27480138271604938271870114918720067827110789528890

Offset: 0

Christian G. Bower, Jan 15 1998, Jun 15 1998, Dec 05 2003

nonn

Index entries for sequences related to groupoids

a(n) = 2*A001424(n) - A001329(n). Cf. A001425.

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, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (sum {d|i*2} (d*s_d))^((i*s_i^2-s_i)/2) * (sum {d|i} (d*s_d))^s_i or {i=j == 0 mod 4} (sum {d|i} (d*s_d))^(i*s_i^2) or {i=j == 2 mod 4} (sum {d|i} (d*s_d))^(i*s_i^2-s_i) * (sum {d|i/2} (d*s_d))^(2*s_i) or {i != j} (sum {d|lcm(i, j, 2)} (d*s_d))^(2*i*j*s_i*s_j/lcm(2*i*j)).

Formula corrected by Sean A. Irvine and Christian G. Bower, Jul 13 2012

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)).

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A001329 Number of nonisomorphic groupoids with n elements.

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A258720 Number of non-self-dual groupoids which are equal to their duals.

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A029850 Number of self-converse groupoids.

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A118542 Number of nonisomorphic groupoids with <= n elements.

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