A023814 -id:A023814 - cp's OEIS frontend

A058166 Triangle read by rows: T(n,k) = number of labeled semigroups of order n with k idempotents.

1, 4, 4, 24, 54, 35, 356, 1044, 1488, 604, 16585, 31620, 60900, 57900, 16727, 3461916, 1699290, 3345420, 4744380, 3128880, 681232, 6058301508, 265521354, 263429355, 439698420, 455785470, 222132666, 38187291

Offset: 1

Christian G. Bower, Nov 15 2000

			1; 4,4; 24,54,35; 356,1044,1488,604; ...

Index entries for sequences related to semigroups

Row sums give A023814.

A351730 Number of labeled idempotent semigroups of order n.

Original entry on oeis.org

1, 1, 4, 35, 604, 16727, 681232, 38187291, 2810370122, 261999605819

Offset: 0

Andrew Howroyd, Feb 17 2022

a(n) is also the number of labeled idempotent monoids of order n+1 with a fixed identity.

Main diagonal of A058166 and A058158.

Cf. A023814, A058112 (isomorphism classes), A351731.

A383871 Number of labeled 3-nilpotent semigroups of order n.

Original entry on oeis.org

0, 0, 6, 180, 11720, 3089250, 5944080072, 147348275209800, 38430603831264883632, 90116197775746464859791750, 2118031078806486819496589635743440, 966490887282837500134221233339527160717340, 17165261053166610940029331024343115375665769316911576, 6444206974822296283920298148689544172139277283018112679406098010

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, A023815.

Cf. A383885, A383886.

a(n) = Sum_{2 <= m <= b(n)} binomial(n,m) * m * Sum_{0 <= i <= m-1} (-1)^i * binomial(m-1,i) * (m-i)^((n-m)^2), where b(n) = floor(n + 1/2 - sqrt(n-3/4)).

A383885 Number of nonisomorphic 3-nilpotent semigroups of order n.

Original entry on oeis.org

0, 0, 1, 9, 118, 4671, 1199989, 3661522792, 105931872028455, 24834563582168716305, 53061406576514239124327751, 2017720196187069550262596208732035, 2756576827989210680367439732667802738773384, 73919858836708511517426763179873538289329852786253510, 29599937964452484359589007277447538854227891149791717673581110642

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, A027851.

Cf. A383871, A383886.

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

For exact formula see the Distler and Mitchell paper.

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.

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

A256411 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ascendingly generated semigroups of order n with k generators.

Original entry on oeis.org

1, 2, 8, 3, 37, 113, 4, 145, 1257, 3492, 5, 452, 9020, 67394, 183732, 6, 1374, 60826, 938194, 6398792, 17061118, 7, 3933, 356023, 30492722, 466578957, 3032145644, 7743056064

Offset: 1

N. J. A. Sloane, Apr 03 2015

			Triangle begins:
  1;
  2,    8;
  3,   37,    113;
  4,  145,   1257,     3492;
  5,  452,   9020,    67394,    183732;
  6, 1374,  60826,   938194,   6398792,   17061118;
  7, 3933, 356023, 30492722, 466578957, 3032145644, 7743056064;
  ...

Alex Bailey, Martin Finn-Sell, Robert Snocken, Subsemigroup, ideal and congruence growth of free semigroups, arXiv preprint arXiv:1409.2444 [math.GR], 2014.

Main diagonal is A023814. Column 2 is A256412.

A084965 Number of labeled totally ordered semigroups with n elements.

Original entry on oeis.org

1, 1, 6, 44, 386, 3852, 42640, 516791, 6817378, 98091071, 1569786228

Offset: 0

Michael Somos, Jun 15 2003

Petr Gajdoš and Martin Kuřil, Ordered semigroups of size at most 7 and linearly ordered semigroups of size at most 10, Semigroup Forum, 89 (2014), 639-663.
J. Slaney, FINDER: Finite Domain Enumerator

Cf. A023814.

a(9)-a(10) from Gajdoš and Kuřil added by Andrey Zabolotskiy, Mar 24 2021

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).

cp's OEIS Frontend

A058166 Triangle read by rows: T(n,k) = number of labeled semigroups of order n with k idempotents.

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A351730 Number of labeled idempotent semigroups of order n.

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A383871 Number of labeled 3-nilpotent semigroups of order n.

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A383885 Number of nonisomorphic 3-nilpotent semigroups of order n.

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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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A118581 Number of nonisomorphic semigroups of order <= n.

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A118601 Partial sums of A058129.

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A256411 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ascendingly generated semigroups of order n with k generators.

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A084965 Number of labeled totally ordered semigroups 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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