A001423 -id:A001423 - 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

A027851 Number of nonisomorphic semigroups of order n.

Original entry on oeis.org

1, 1, 5, 24, 188, 1915, 28634, 1627672, 3684030417, 105978177936292

Offset: 0

Christian G. Bower, Dec 13 1997, updated Feb 19 2001

Özgur Akgun, Mun See Chang, Ian P. Gent, and Christopher Jefferson, Breaking the Symmetries of Indistinguishable Objects, arXiv:2503.17251 [cs.AI], 2025. See pp. 14, 17.
Peter Cameron's Blog, The combinatorial explosion, Posted 18/02/2016.
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.
C. Noebauer, Home page
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
Arman Shamsgovara, Enumerating, Cataloguing and Classifying All Quantales on up to Nine Elements, In: Glück, R., Santocanale, L., and Winter, M. (eds), Relational and Algebraic Methods in Computer Science (RAMiCS 2023) Lecture Notes in Computer Science, Springer, Cham, Vol. 13896.
Jeremy G. Sumner, Michael D. Woodhams, Lie-Markov models derived from finite semigroups, arXiv:1709.00520 [math.GR], 2017.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).
A. de Vries, Formal Languages: An Introduction, 2012.
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Cf. A001426, A023814, A058108.

Cf. A001423, A029851, A079173, A001329.

a(n) = A001423(n)*2 - A029851(n).

a(n) + A079173(n) = A001329(n).

a(8)-a(9) from Andreas Distler, Jan 13 2011

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

A001425 Number of commutative groupoids with n elements.

Original entry on oeis.org

1, 1, 4, 129, 43968, 254429900, 30468670170912, 91267244789189735259, 8048575431238519331999571800, 24051927835861852500932966021650993560, 2755731922430783367615449408031031255131879354330

Offset: 0

N. J. A. Sloane

nonn

Satoh, S.; Yama, K.; and Tokizawa, M., Semigroups of order 8, Semigroup Forum 49 (1994), 7-29. [Background]
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).
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.

Eric Postpischil Posting to sci.math newsgroup, May 21 1990
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)
Index entries for sequences related to groupoids

a(n)+A079183(n)=A001329(n)

Cf. A001329, A023813, A038016.

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} (d*s_d))^((i*s_i^2+s_i)/2) or {i=j, even} (sum {d|i} (d*s_d))^(i*s_i^2/2) * (sum {d|i/2} (d*s_d))^s_i or {i != j} (sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j)

a(n) asymptotic to (n^binomial(n+1, 2))/n! = A023813(n)/A000142(n) ~ e^n*n^binomial(n, 2) / sqrt(2*pi*n).

More terms from Christian G. Bower Feb 15 1998 and May 15 1998. Formula Dec 03 2003.

A001428 Number of inverse 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, 2, 5, 16, 52, 208, 911, 4637, 26422, 169163, 1198651, 9324047, 78860687, 719606005, 7035514642

Offset: 1

N. J. A. Sloane

S. Satoh, K. Yama, and M. Tokizawa, Semigroups of order 8, Semigroup Forum 49 (1994), 7-29.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, Semigroup Forum, 14 (1977), 69-79.
R. J. Plemmons, Cayley Tables for All Semigroups of Order Less Than 7. Department of Mathematics, Auburn Univ., 1965.
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).
M. V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific, 1998. [From Jonathan Vos Post, Mar 08 2010]
G. B. Preston, "Inverse semi-groups". Journal of the London Mathematical Society 29: 396-403. [From Jonathan Vos Post, Mar 08 2010]
V. V. Wagner (1952). "Generalised groups". Proceedings of the USSR Academy of Sciences 84: 1119-1122. (Russian) English translation. [From Jonathan Vos Post, Mar 08 2010]

Joao Araujo and Michael Kinyon, An elegant 3-basis for inverse semigroups, March 21, 2010. [From _Jonathan Vos Post_, Mar 23 2010]
Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Luna Elliott, Alex Levine, and James Mitchell, E-disjunctive inverse semigroups, arXiv:2405.19825 [math.GR], 2024. See p. 3.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, annotated and scanned copy.
Martin E. Malandro, Enumeration of finite inverse semigroups, arXiv:1312.7192 [math.CO]
R. J. Plemmons, There are 15973 semigroups of order 6 (annotated and scanned copy)
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)
Wikipedia, Inverse semigroup
Index entries for sequences related to semigroups

Cf. A234843 (commutative inverse semigroups), A234844 (inverse monoids), A234845 (commutative inverse monoids).

a(8) and a(9) from Andreas Distler, Jan 17 2011

Added more terms (from the Malandro reference), Joerg Arndt, Dec 30 2013

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

A258719 Number of self-dual noncommutative groupoids with n elements.

Original entry on oeis.org

0, 0, 9, 16192, 198862625, 42002510818752, 207278622607612079818, 29215384735442091573649485568, 137562588659577384442574662095693261747, 24724406349174154904254665510689036571978910174560

Offset: 1

N. J. A. Sloane, Jun 18 2015

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. A029850, A001425, A258720.

a(n) = A029850(n) - A001425(n). - Andrew Howroyd, May 06 2023

a(5)-a(10) from Andrew Howroyd, May 06 2023

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

A058133 Number of monoids (semigroups with identity) of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

0, 1, 2, 6, 27, 156, 1373, 17730, 858977, 1844075697, 52991253973742

Offset: 0

Christian G. Bower, Nov 13 2000

Eric Postpischil Posting to sci.math newsgroup, May 21 1990
Andreas Distler and Tom Kelsey, The Monoids of Order Eight and Nine. Lecture Notes in Computer Science 5144 (2008), 61-76.
Andreas Distler and Tom Kelsey, The Monoids of Orders Eight, Nine & Ten. Annals of Mathematics and Artificial Intelligence 56 (2009), 3-25.
Index entries for sequences related to monoids

Cf. A001423, A058129, A058132, A151823.

a(n)=(A058129(n)+A058132(n))/2.

a(8) and a(9) from Distler & Kelsey (2008), added by N. J. A. Sloane, Jul 10 2009
a(10) from Distler & Kelsey (2009), added by Max Alekseyev, Jul 13 2009
a(0) prepended by Jianing Song, Oct 26 2019

cp's OEIS Frontend

A001329 Number of nonisomorphic groupoids with n elements.

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A027851 Number of nonisomorphic semigroups of order n.

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A001426 Number of commutative semigroups of order n.

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

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A001425 Number of commutative groupoids with n elements.

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A001428 Number of inverse semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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

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A258719 Number of self-dual noncommutative 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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A058133 Number of monoids (semigroups with identity) of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).