A001423 -id:A001423 - cp's OEIS frontend

A029851 Number of self-converse semigroups of order n.

Original entry on oeis.org

1, 1, 3, 12, 64, 405, 3312, 44370, 2209839, 623492664

Offset: 0

Christian G. Bower, Jan 27 1998, updated Feb 19 2001

Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Eric Weisstein's World of Mathematics, SemiGroup.
Index entries for sequences related to semigroups

Cf. A001423, A001426, A027851.

a(n) = A001423(n)*2 - A027851(n).

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

A058123 Triangle read by rows: semigroups of order n with k idempotents, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

1, 2, 2, 5, 7, 6, 19, 37, 44, 26, 132, 216, 351, 326, 135, 3107, 1780, 3093, 4157, 2961, 875, 623615, 32652, 33445, 53145, 56020, 30395, 6749, 1834861133, 4665709, 600027, 754315, 1007475, 822176, 348692, 60601, 52976551026562, 12710266442, 68769167, 14050493, 18660074, 20044250, 12889961, 4389418, 618111

Offset: 1

Christian G. Bower, Nov 10 2000

			Triangle starts:
    1;
    2,   2;
    5,   7,   6;
   19,  37,  44,  26;
  132, 216, 351, 326, 135;
  ...

Andrey Zabolotskiy, Table of n, a(n) for n = 1..55 (rows 1-10)
Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Andreas Distler, Chris Jefferson, Tom Kelsey, 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.
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 A001423. Main diagonal: A002788. Columns 1-3: A002786, A002787, A005591.

More terms from Andreas Distler, Jan 13 2011

A001424 Number of nonisomorphic and nonantiisomorphic groupoids with n elements.

Original entry on oeis.org

1, 1, 7, 1734, 89521056, 1241763995193675, 7162795001695681351632672, 25488450150907292192918677242007992558, 77841043345568973636021269757801814299054870565039692

Offset: 0

N. J. A. Sloane

nonn

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)
Eric Weisstein's World of Mathematics, Groupoid.
Index entries for sequences related to groupoids

Cf. A001329, A029850.

a(n) = (A001329(n) + A029850(n))/2

Better description and corrected 4th term from Christian G. Bower, Jan 15 1998. More terms, Jun 15 1998.

A151823 Number of nonequivalent monoids of order n with more than one invertible element.

Original entry on oeis.org

0, 1, 2, 9, 30, 213, 1757, 22956, 955569, 1853259264

Offset: 1

N. J. A. Sloane, Jul 10 2009

nonn

a(n) is also the number of nonequivalent monoids with nontrivial unit group. [From Tom Kelsey (tom(AT)cs.st-and.ac.uk), Apr 01 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, pp. 61-76, Springer-Verlag.
Andreas Distler and Tom Kelsey, The Monoids of Orders Eight, Nine & Ten, Annals of Mathematics and Artificial Intelligence, 56 (2009), 3-25. [From Tom Kelsey (tom(AT)cs.st-and.ac.uk), Apr 01 2010]

Cf. A001423, A058133.

Definition corrected by Tom Kelsey (tom(AT)cs.st-andrews.ac.uk), Apr 01 2010

Corrected and extended by Tom Kelsey (tom(AT)cs.st-and.ac.uk), Apr 01 2010

A002788 Idempotent 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, 2, 6, 26, 135, 875, 6749, 60601, 618111, 7033090

Offset: 0

N. J. A. Sloane

An idempotent semigroup is one whose elements are all idempotents.

R. J. Plemmons, There are 15973 semigroups of order 6, Math. Algor., 2 (1967), 2-17; 3 (1968), 23.
R. J. Plemmons, Construction and analysis of non-equivalent finite semigroups, pp. 223-228 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
S. Satoh, K. Yama and M. Tokizawa, Semigroups of order 8; Semigroup Forum 49, 1994.
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).

Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
R. J. Plemmons, There are 15973 semigroups of order 6 (annotated and scanned copy)
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Cf. A001423. Main diagonal of A058123.

Additional reference and comments from Michael Somos
a(7) term from Christian G. Bower, Feb 19 2001
a(8) (from the Satoh et al. reference) sent by Tom Kelsey (tom(AT)cs.st-and.ac.uk), Jun 17 2008
a(9)-a(10) from Andreas Distler, Jan 12 2011

A058107 Number of asymmetric 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, 3, 12, 78, 746, 10965, 746277

Offset: 0

Christian G. Bower, Nov 09 2000

Index entries for sequences related to semigroups

Cf. A001423.

Updated Feb 19 2001

A383886 Number of 3-nilpotent semigroups, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

0, 0, 1, 8, 84, 2660, 609797, 1831687022, 52966239062973, 12417282095522918811, 26530703289252298687053072, 1008860098093547692911901804990610, 1378288413994605341053354105969660808031163, 36959929418354255758713676933402538920157765946956889, 14799968982226242179794503639146983952853044950740907666303436922

Offset: 1

Elijah Beregovsky, May 13 2025

nonn

A semigroup S is nilpotent if there exists a natural number r such that the set S^r of all products of r elements of S has size 1.
If r is the smallest such number, then S is said to have nilpotency degree r.
This sequence counts semigroups S that have an element e such that for all x,y,z in S x*y*z = e.
In 1976 Kleitman, Rothschild and Spencer gave an argument asserting that the proportion of 3-nilpotent semigroups, amongst all semigroups of order n, is asymptotically 1. Later opinion regards their argument as incomplete, and no satisfactory proof has been found.

H. Jürgensen, F. Migliorini, and J. Szép, Semigroups. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1991.

Andreas Distler and James D. Mitchell, The number of nilpotent semigroups of degree 3, arXiv:1201.3529 [math.CO], 2012.
Igor Dolinka, D. G. FitzGerald, and James D. Mitchell, Semirigidity and the enumeration of nilpotent semigroups of index three, arXiv:2411.00466 [math.CO], 2024.
Pierre A. Grillet, Counting Semigroups, Communications in Algebra, 43(2), 574-596, (2014).
D. J. Kleitman, B. R. Rothschild, and J. H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc. 55 (1976), 227-232.
Index entries for sequences related to semigroups

Cf. A023814, A001423.

Cf. A383885, A383871.

a(n) = A383871(n)/2n! * (1+o(1)). See Grillet paper in Links.

A002786 Semigroups of order n with 1 idempotent, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

1, 2, 5, 19, 132, 3107, 623615, 1834861133, 52976551026562, 12417619575092896741

Offset: 1

N. J. A. Sloane

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).

Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Andreas Distler, Chris Jefferson, Tom Kelsey, 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. a(10) is the sum of entries of Tables 4 and 5; note that Table 4 has incorrect Total.
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 Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Column 1 of A058123.

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

a(10) from Andrey Zabolotskiy, Nov 08 2018

A002787 Number of semigroups of order n with 2 idempotents, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

2, 7, 37, 216, 1780, 32652, 4665709, 12710266442, 381279977009776

Offset: 2

N. J. A. Sloane

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).

Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Andreas Distler, Chris Jefferson, Tom Kelsey, 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. a(10) is the Table 3 Total.
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 Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Column 2 of A058123.

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

a(10) from Andrey Zabolotskiy, Nov 08 2018

A005591 Number of semigroups of order n with 3 idempotents, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

6, 44, 351, 3093, 33445, 600027, 68769167, 219587421825

Offset: 3

N. J. A. Sloane

nonn

R. J. Plemmons, There are 15973 semigroups of order 6, Math. Algor., 2 (1967), 2-17; 3 (1968), 23.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Andreas Distler, Chris Jefferson, Tom Kelsey, 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. a(10) is at the top row of Table 2.
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.
Index entries for sequences related to semigroups

Column 3 of A058123.

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

a(10) from Andrey Zabolotskiy, Nov 08 2018

cp's OEIS Frontend

A029851 Number of self-converse semigroups of order n.

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A058123 Triangle read by rows: semigroups of order n with k idempotents, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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A001424 Number of nonisomorphic and nonantiisomorphic groupoids with n elements.

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A151823 Number of nonequivalent monoids of order n with more than one invertible element.

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A002788 Idempotent semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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A058107 Number of asymmetric semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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A383886 Number of 3-nilpotent semigroups, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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A002786 Semigroups of order n with 1 idempotent, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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A002787 Number of semigroups of order n with 2 idempotents, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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A005591 Number of semigroups of order n with 3 idempotents, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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