A027851 -id:A027851 - 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

A001423 Number of semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

1, 1, 4, 18, 126, 1160, 15973, 836021, 1843120128, 52989400714478, 12418001077381302684

Offset: 0

N. J. A. Sloane

David Nacin, "Puzzles, Parity Maps, and Plenty of Solutions", Chapter 15, The Mathematics of Various Entertaining Subjects: Volume 3 (2019), Jennifer Beineke & Jason Rosenhouse, eds. Princeton University Press, Princeton and Oxford, p. 245.
R. J. Plemmons, There are 15973 semigroups of order 6, Math. Algor., 2 (1967), 2-17; 3 (1968), 23.
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).

A. de Vries, Formal Languages: An Introduction
Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Andreas Distler and Tom Kelsey, The Monoids of Order Eight and Nine, in Intelligent Computer Mathematics, Lecture Notes in Computer Science, Volume 5144/2008, Springer-Verlag. [From _N. J. A. Sloane_, Jul 10 2009]
A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013.
Andreas Distler, Chris Jefferson, Tom Kelsey, and Lars Kotthoff, The Semigroups of Order 10, in: M. Milano (Ed.), Principles and Practice of Constraint Programming, 18th International Conference, CP 2012, Québec City, QC, Canada, October 8-12, 2012, Proceedings (LNCS, volume 7514), pp. 883-899, Springer-Verlag Berlin Heidelberg 2012.
Remigiusz Durka and Kamil Grela, On the number of possible resonant algebras, arXiv:1911.12814 [hep-th], 2019.
G. E. Forsythe, SWAC computes 126 distinct semigroups of order 4, Proc. Amer. Math. Soc. 6, (1955). 443-447.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, Semigroup Forum, 14 (1977), 69-79.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, annotated and scanned copy.
Daniel J. Kleitman, Bruce L. Rothschild and Joel H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc., 55 (1976), 227-232.
R. J. Plemmons, There are 15973 semigroups of order 6 (annotated and scanned copy)
Eric Postpischil Associativity Problem, Posting to sci.math newsgroup, May 21 1990.
S. Satoh, K. Yama, and M. Tokizawa, Semigroups of order 8, Semigroup Forum 49 (1994), 7-29.
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)
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Cf. A001426, A023814, A058107, A058123, A151823.

a(n) = (A027851(n) + A029851(n))/2.

a(9) added by Andreas Distler, Jan 12 2011

a(10) from Distler et al. 2012, added by Andrey Zabolotskiy, Nov 08 2018

A001426 Number of commutative semigroups of order n.

Original entry on oeis.org

1, 1, 3, 12, 58, 325, 2143, 17291, 221805, 11545843, 3518930337

Offset: 0

N. J. A. Sloane

P. A. Grillet, Computing Finite Commutative Semigroups, Semigroup Forum 53 (1996), 140-154.
P. A. Grillet, Computing Finite Commutative Semigroups: Part II, Semigroup Forum 67 (2003), 159-184.
R. J. Plemmons, There are 15973 semigroups of order 6, Math. Algor., 2 (1967), 2-17; 3 (1968), 23.
R. J. Plemmons, Cayley Tables for All Semigroups of Order Less Than 7. Department of Mathematics, Auburn Univ., 1965.
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).

Remigiusz Durka, Kamil Grela, On the number of possible resonant algebras, arXiv:1911.12814 [hep-th], 2019.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, Semigroup Forum, 14 (1977), 69-79.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, annotated and scanned copy.
R. J. Plemmons, There are 15973 semigroups of order 6 (annotated and scanned copy)
Eric Postpischil Posting to sci.math newsgroup, May 21 1990 [Broken link]
S. Satoh, K. Yama, M. Tokizawa, Semigroups of order 8, Semigroup Forum 49 (1994), 7-29.
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)
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Cf. A001423, A023815, A027851, A058105, A058116.

a(n) + A079193(n) + A079196(n) + A079199(n) = A001329(n).

a(8) (from the Satoh et al. paper) supplied by Richard C. Schroeppel, Jul 22 2005

a(9) and a(10) from Grillet references sent by Jens Zumbragel (jzumbr(AT)math.unizh.ch), Jun 14 2006

A023814 Number of associative binary operations on an n-set; number of labeled semigroups.

Original entry on oeis.org

1, 1, 8, 113, 3492, 183732, 17061118, 7743056064, 148195347518186, 38447365355811944462

Offset: 0

Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Alex Bailey, Martin Finn-Sell, and Robert Snocken, Subsemigroup, ideal and congruence growth of free semigroups, arXiv preprint arXiv:1409.2444 [math.GR], 2014.
A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013.
C. Noebauer, Home page [broken link]
C. Noebauer, The Numbers of Small Rings
C. Noebauer, Thesis on the enumeration of near-rings
Eric Postpischil Posting to sci.math newsgroup, May 21 1990
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Cf. A001423, A001426, A023813, A023815, A027851.

a(n) + A079172(n) = A002489(n).

a(8), a(9) from Distler and Kelsey (2013). - N. J. A. Sloane, Feb 19 2013

A058129 Number of nonisomorphic monoids (semigroups with identity) of order n.

Original entry on oeis.org

0, 1, 2, 7, 35, 228, 2237, 31559, 1668997, 3685886630

Offset: 0

Christian G. Bower, Nov 13 2000

Geoff Cruttwell, Counting Finite Categories, presentation, (2018).
Remigiusz Durka and Kamil Grela, On the number of possible resonant algebras, arXiv:1911.12814 [hep-th], 2019.
Najwa Ghannoum, Investigation of finite categories, Doctoral thesis, Univ. Côte d'Azur (France); Univ. Libanaise (Lebanon), tel-0394832 [math.CT], 2022.
Pierre A. Grillet, Counting Semigroups, Communications in Algebra, 43(2), 574-596, (2014).
Mikhail Kornev, On the Classification of n-Valued Monoids and Groups of Order 3, arXiv:2508.04454 [math.GR], 2025. See p. 11.
Václav Koubek and Vojtěch Rödl, Note on the number of monoids of order n, Commentationes Mathematicae Universitatis Carolinae 026.2 (1985): 309-314.
Eric Postpischil, Posting to sci.math newsgroup, May 21 1990
Clayton Cristiano Silva, Irreducible Numerical Semigroups, University of Campinas, São Paulo, Brazil (2019).
Index entries for sequences related to monoids

Cf. A058132, A058133, A058153.

Cf. A027851 (number of all nonisomorphic semigroups).

a(n) = 2*A058133(n) - A058132(n).
a(n) < A027851(n) except for equality iff n = 1. - M. F. Hasler, Dec 10 2018
From Elijah Beregovsky, May 13 2025 (Start):
a(n) >= A027851(n-1).
Conjecture: a(n) = A027851(n-1)*(1+o(1)). See Koubek and Rödl paper in the Links.
Conjecture: a(n) = A058153(n)/n! * (1+o(1)). See Grillet paper in the Links. (End)

a(8) from Christian G. Bower, Dec 26 2006
a(0) = 0 prepended by M. F. Hasler, Dec 10 2018
a(9) from Elijah Beregovsky, from the work of G. Cruttwell and R. Leblanc, May 12 2025

A023815 Number of binary operations on an n-set that are commutative and associative; labeled commutative semigroups.

Original entry on oeis.org

1, 1, 6, 63, 1140, 30730, 1185072, 66363206, 7150843144, 3829117403448

Offset: 0

Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Eric Postpischil Posting to sci.math newsgroup, May 21 1990
Amit Sehgal, Sunil Kumar, Sarita, Yashpal, Commutative Associative Binary Operations on a Set with Five Elements, J. Phys.: Conf. Ser. (2018) 1000 012063
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Row sums of A058167.
Cf. A001423, A001426 (isomorphism classes), A023813 (commutative only), A023814 (associative only), A027851.
Cf. A002489, A079192, A079195, A079198, A079201, A079210.

a(n) + A079192(n) + A079195(n) + A079198(n) = A002489(n).

a(n) = Sum_{k>=1} A079201(n,k)*A079210(n,k). - Andrew Howroyd, Jan 26 2022

a(8) from Andrew Howroyd, Jan 26 2022

a(9) from Andrew Howroyd, Feb 14 2022

A079199 Number of isomorphism classes of associative non-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 2, 12, 130, 1590, 26491, 1610381, 3683808612, 105978166390449

Offset: 1

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

nonn

A079193(n)+A079196(n)+A079199(n)+A001426(n)=A001329(n).
Each a(n) is equal to the sum of the elements in row n of A079200.
Since this is the number of nonisomorphic noncommutative semigroups of order n, A079199(n)=A027851(n)-A001426(n). - Stanislav Sykora, Apr 03 2016

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

Cf. A001426, A027851, A079193, A079196, A079198, A079200.

Added terms a(5)-a(9). - Stanislav Sykora, Apr 03 2016

A058108 Triangle read by rows: number of nonisomorphic semigroups of order n with k idempotents.

Original entry on oeis.org

1, 2, 3, 5, 9, 10, 20, 50, 72, 46, 171, 309, 590, 594, 251, 5284, 2806, 5422, 7772, 5668, 1682, 1224331, 58583, 61323, 101539, 109107, 59576, 13213, 3667785000, 9207430, 1150085, 1466691, 1983558, 1626956, 690871, 119826, 105952488687468, 25412267163, 136799017, 27690828, 36991211, 39865274, 25666762, 8739857, 1228712

Offset: 1

Christian G. Bower, Nov 09 2000

			Triangle begins:
    1;
    2,   3;
    5,   9,  10;
   20,  50,  72,  46;
  171, 309, 590, 594, 251;
  ...

Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013.
Index entries for sequences related to semigroups

Row sums give A027851. Main diagonal: A058112. Columns 1-3: A058109-A058111.

Updated Feb 19 2001

More terms from Andreas Distler, Jan 13 2011

A079193 Number of isomorphism classes of non-associative non-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 4, 3189, 178937854, 2483527282663335, 14325590003288422852078277, 50976900301814584087291456618542388746

Offset: 1

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

nonn

A079193(n)+A079196(n)+A079199(n)+A001426(n)=A001329(n).

Each a(n) is equal to the sum of the elements in row n of A079194.

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

Cf. A079192, A079194, A079196, A079199, A001426.
a(n) = A001329(n)-A001425(n)-A027851(n)+A001426(n).
a(n)+A079196(n)+A079199(n)+A001426(n)=A001329(n).

Edited and extended by Christian G. Bower, Nov 26 2003

A079173 Number of isomorphism classes of non-associative closed binary operations on a set of order n.

Original entry on oeis.org

0, 5, 3306, 178981764, 2483527537092910, 14325590003318891522247046, 50976900301814584087291487087212542367, 155682086691137947272042502251643461917498831796991599, 541851439802559836957713164869818405872834954135521300809796661279574643

Offset: 1

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

nonn

Each a(n) is equal to the sum of the elements in row n of A079174.

Christian van den Bosch, Closed binary operations on small sets. [Wayback Machine link]
Index entries for sequences related to groupoids.

Cf. A001329, A027851, A079172, A079174.

a(n) = A001329(n) - A027851(n).

More terms from Christian G. Bower, Nov 26 2003

a(8)-a(9) added using the data at A001329 and A027851 by Amiram Eldar, Jul 19 2025

cp's OEIS Frontend

A001329 Number of nonisomorphic groupoids with n elements.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A001423 Number of semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A001426 Number of commutative semigroups of order n.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A023814 Number of associative binary operations on an n-set; number of labeled semigroups.

Views

Author

Keywords

Links

Crossrefs

Extensions

A058129 Number of nonisomorphic monoids (semigroups with identity) of order n.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A023815 Number of binary operations on an n-set that are commutative and associative; labeled commutative semigroups.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A079199 Number of isomorphism classes of associative non-commutative closed binary operations on a set of order n.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A058108 Triangle read by rows: number of nonisomorphic semigroups of order n with k idempotents.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A079193 Number of isomorphism classes of non-associative non-commutative closed binary operations on a set of order n.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A079173 Number of isomorphism classes of non-associative closed binary operations on a set of order n.