A001426 -id:A001426 - 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

A058116 Triangle read by rows: T(n,k) is the number of isomorphism classes of commutative semigroups of order n with k idempotents.

Original entry on oeis.org

1, 2, 1, 5, 5, 2, 16, 23, 14, 5, 62, 106, 93, 49, 15, 342, 544, 582, 422, 200, 53, 3435, 3380, 3773, 3360, 2178, 943, 222, 97061, 30788, 27222, 26625, 21283, 12676, 5072, 1078

Offset: 1

Christian G. Bower, Nov 09 2000

			Triangle begins:
   1;
   2,   1;
   5,   5,  2;
  16,  23, 14,  5;
  62, 106, 93, 49, 15;
  ...

Index entries for sequences related to semigroups

Row sums give A001426.
Main diagonal is A006966(n+1).
Column 1 is A058117.
The labeled version is A058167.
Cf. A058108, A058142.

a(29)-a(36) from Andrew Howroyd, Jan 27 2022

A058105 Number of commutative asymmetric semigroups of order n.

Original entry on oeis.org

1, 1, 3, 9, 39, 201, 1258, 10013, 142755

Offset: 0

Christian G. Bower, Nov 09 2000

Index entries for sequences related to semigroups

Cf. A001426, A079201.

a(n) = A079201(n, A027423(n)).

Updated Feb 19 2001.
a(7) corrected Dec 26 2006
a(8) from Andrew Howroyd, Jan 26 2022

A058131 Number of isomorphism classes of commutative monoids (commutative semigroups with identity) of order n.

Original entry on oeis.org

0, 1, 2, 5, 19, 78, 421, 2637, 20486, 246458, 11833361

Offset: 0

Christian G. Bower, Nov 13 2000

Remigiusz Durka and Kamil Grela, On the number of possible resonant algebras, arXiv:1911.12814 [hep-th], 2019.
Eric Postpischil, Posting to sci.math newsgroup, May 21 1990
Eric Weisstein's World of Mathematics, Commutative Monoid
Eric Weisstein's World of Mathematics, Additive Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 952.
Index entries for sequences related to monoids

Row sums of A058142.

Cf. A001426, A058129, A058155 (labeled).

a(0) prepended by Jianing Song, Oct 26 2019
a(8) and a(9) from Alex Meiburg, Oct 20 2021
a(10) from Andrew Howroyd, Feb 15 2022

A118581 Number of nonisomorphic semigroups of order <= n.

Original entry on oeis.org

1, 2, 7, 31, 219, 2134, 30768, 1658440, 3685688857, 105981863625149

Offset: 0

Jonathan Vos Post, May 07 2006

Semigroup analog of A063756 Number of groups of order <= n. a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation (and thus is an associative groupoid). Some sources require that a semigroup have an identity element (in which case semigroups are identical to monoids). Not all sources agree that S should be nonempty. This sequence assumes that a semigroup may be empty and need not have an identity.

			a(7) = 1658440 = 1 + 1 + 5 + 24 + 188 + 1915 + 28634 + 1627672.

Cf. A001329, A001423, A001426, A023814, A027851, A029851, A058108, A063756, A079173.

a(n) = Sum_{i=0..n} A027851(i). a(n) = Sum_{i=0..n} (2*A001423(i) - A029851(i)).

a(8)-a(9) (using A027851) from Giovanni Resta, Jun 16 2016

A118601 Partial sums of A058129.

Original entry on oeis.org

1, 3, 10, 45, 273, 2510, 34069, 1703066

Offset: 1

Jonathan Vos Post, May 08 2006

Cf. A001329, A001423, A001426, A023814, A027851, A029851, A058108, A058132, A058133, A063756, A079173, A118581.

a(n) = SUM[i=1..n] A058129(i). a(n) = SUM[i=1..n] (2*A058133(i) - A058132(i)).

One more term from Jonathan Vos Post, Jul 20 2009

Edited by N. J. A. Sloane, Jul 25 2009

A118100 Number of commutative semigroups of order <= n.

Original entry on oeis.org

1, 2, 5, 17, 75, 400, 2543, 19834, 241639, 11787482, 3530717819

Offset: 0

Jonathan Vos Post, May 11 2006

A001426(n) is the number of commutative semigroups of order n. A001426(n) + A079193(n) + A079196(n) + A079199(n) = A001329(n). 2, 5, 17, 2543 and 241639 are primes.

			a(8) = 1 + 1 + 3 + 12 + 58 + 325 + 2143 + 17291 + 221805 = 241639.

Index entries for sequences related to semigroups.

Cf. A001423, A001426, A023815, A027851, A058105, A058116.

a(n) = Sum_{i=1..n} A001426(i).

a(9)-a(10) added using the terms in A001426 by Miles Englezou, May 27 2025

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)).

A186117 Number of nonisomorphic semigroups of order n minus number of groups of order n.

Original entry on oeis.org

0, 4, 23, 186, 1914, 28632, 1627671, 3684030412, 105978177936290

Offset: 1

Jonathan Vos Post, Feb 13 2011

In a sense, this measures the increase in combinatorial structures available by dropping the requirement of inverses, and an identity element, in moving from the group axioms to the semigroup axioms. A semigroup is mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. Other sequences may be derived by considering commutative semigroups and commutative groups, self-converse semigroup, counting idempotents, and the like.

			a(1) = 0 because there are unique groups and semigroups of order 1, so 1 - 1  = 0.
a(2) = 4 because there are 5 semigroups of order 2 groups and a unique group of order 2, so 5 - 1  = 4.

Eric W. Weisstein, Finite Group
Eric W. Weisstein, Semigroup

Cf. A000001, A027851, A001423, A029851, A001426, A023814, A058108, A079173, A001329, A186116.

a(n) = A027851(n) - A000001(n).

A328746 Number of loops 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, 1, 1, 2, 5, 72, 12151, 53146457

Offset: 0

Jianing Song, Oct 26 2019

For the number of group-like algebraic structures of order n, see:
Semigroups: A027851 or A001423 (commutative: A001426);
Monoids: A058129 or A058133 (commutative: A058131);
Quasigroups: A057991 or A058171 (commutative: A057992);
Loops: A057771 or this sequence (commutative: A089925);
Groups: A000001 (commutative: A000688);
Rings: A027623 or A038036 (commutative: A037289);
Rings with unity: A037291;
Fields: A069513.

a(n) = (A057771(n)+A057996(n))/2.

cp's OEIS Frontend

A029851 Number of self-converse semigroups of order n.

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A058116 Triangle read by rows: T(n,k) is the number of isomorphism classes of commutative semigroups of order n with k idempotents.

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A058105 Number of commutative asymmetric semigroups of order n.

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A058131 Number of isomorphism classes of commutative monoids (commutative semigroups with identity) of order n.

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A118581 Number of nonisomorphic semigroups of order <= n.

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A118601 Partial sums of A058129.

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A118100 Number of commutative semigroups of order <= n.

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A118542 Number of nonisomorphic groupoids with <= n elements.

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A186117 Number of nonisomorphic semigroups of order n minus number of groups of order n.

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A328746 Number of loops of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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