A058116 -id:A058116 - cp's OEIS frontend

A006966 Number of lattices on n unlabeled nodes.

1, 1, 1, 1, 2, 5, 15, 53, 222, 1078, 5994, 37622, 262776, 2018305, 16873364, 152233518, 1471613387, 15150569446, 165269824761, 1901910625578, 23003059864006

Offset: 0

N. J. A. Sloane

Also commutative idempotent monoids. Also commutative idempotent semigroups of order n-1.

Commutative idempotent semigroups are also called semilattices, so A(n) counts semilattices of order n-1. - Dennis Sweeney, Jul 19 2024

J. Heitzig and J. Reinhold, Counting finite lattices, Algebra Universalis, 48 (2002), 43-53.
P. D. Lincoln, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. R. Stembridge, personal communication.

David Wasserman and Nathan Lawless, Table of n, a(n) for n = 0..20 (a(20) from _Volker Gebhardt_)
R. Belohlavek and V. Vychodil, Residuated lattices of size <=12, Order 27 (2010) 147-161 doi:10.1007/s11083-010-9143-7, Table 2.
V. Gebhardt and S. Tawn, Constructing unlabelled lattices, arXiv:1609.08255 [math.CO], 2016.
D. J. Greenhoe, MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing, 2014.
J. Heitzig and J. Reinhold, Counting finite lattices, preprint no. 298, Institut für Mathematik, Universität Hannover, Germany, 1999.
J. Heitzig and J. Reinhold, Counting finite lattices, CiteSeer 1999.
P. Jipsen and N. Lawless, Generating all modular lattices of a given size (preprint)
D. J. Kleitman and K. J. Winston, The asymptotic number of lattices, in: Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978), Ann. Discrete Math. 6 (1980), 243-249.
S. Kyuno, An inductive algorithm to construct finite lattices, Math. Comp. 33 (1979), no. 145, 409-421.
N. Lawless, Generating all modular lattices of a given size, Slides, ADAM 2013.
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.
Wikipedia, Semilattice.
Richard Stanley, MathOverflow: Semilattices with n elements
Index entries for sequences related to semigroups
Index entries for "core" sequences

Cf. A006981, A006982, A055512. Main diagonal of A058142. a(n+1) is main diagonal of A058116.

More terms from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000
a(19) from Nathan Lawless, Sep 15 2013
a(20) from Volker Gebhardt, Sep 28 2016

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

A058142 Triangle read by rows: number of commutative monoids of order n with k idempotents.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 2, 9, 6, 2, 1, 26, 30, 16, 5, 1, 98, 142, 111, 54, 15, 1, 455, 718, 713, 482, 215, 53, 3, 4018, 4277, 4637, 3919, 2414, 996, 222, 2, 101910, 36124, 32937, 31308, 24047, 13758, 5294, 1078, 1, 10054303, 708355, 290278, 260758, 229411, 164216, 88178, 31867, 5994

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  2,  9,   6,   2;
  1, 26,  30,  16, 5;
  1, 98, 142, 111, 54, 15;
  ...

Index entries for sequences related to monoids

Row sums give A058131.
Main diagonal: A006966.
Columns 1..2: A000688, A058143.
Cf. A058116, A058137, A058150, A058159, A058160.

a(30)-a(55) from Andrew Howroyd, Feb 15 2022

A058167 Triangle read by rows: T(n,k) is the number of labeled commutative semigroups of order n with k idempotents.

Original entry on oeis.org

1, 4, 2, 24, 30, 9, 260, 492, 312, 76, 4805, 10060, 9900, 4900, 1065, 157956, 284130, 348420, 259500, 112500, 22566, 12277440, 11892846, 14768775, 14093380, 9063600, 3592554, 674611, 3287166928, 896150920, 812261856, 854806120, 707722680, 413149464, 152565280, 27019896

Offset: 1

Christian G. Bower, Nov 15 2000

			Triangle begins:
     1;
     4,     2;
    24,    30,    9;
   260,   492,  312,   76;
  4805, 10060, 9900, 4900, 1065;
  ...

Index entries for sequences related to semigroups

Row sums give A023815.
Main diagonal is A058164(n+1).
Cf. A058116 (isomorphism classes).

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

A058117 Number of commutative semigroups of order n with 1 idempotent.

Original entry on oeis.org

1, 2, 5, 16, 62, 342, 3435

Offset: 1

Christian G. Bower, Nov 09 2000

Index entries for sequences related to semigroups

Column 1 of A058116.

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

cp's OEIS Frontend

A006966 Number of lattices on n unlabeled nodes.

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A001426 Number of commutative semigroups of order n.

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A058142 Triangle read by rows: number of commutative monoids of order n with k idempotents.

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A058167 Triangle read by rows: T(n,k) is the number of labeled commutative semigroups of order n with k idempotents.

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A058117 Number of commutative semigroups of order n with 1 idempotent.

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A118100 Number of commutative semigroups of order <= n.

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