A058142 -id:A058142 - 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

A058159 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents.

Original entry on oeis.org

1, 2, 2, 3, 18, 6, 16, 180, 144, 36, 30, 2040, 3240, 1740, 380, 360, 43170, 81000, 70740, 31680, 6390, 840, 1400112, 2589510, 2976960, 2055480, 832230, 157962, 15360, 110488616, 117733728, 144285960, 130781280, 79626120, 30004128, 5396888, 68040, 30647444544, 9223088112, 8744866704, 8997002280, 7154708400, 4005012816, 1421659512, 243179064

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
    1;
    2,     2;
    3,    18,     6;
   16,   180,   144,    36;
   30,  2040,  3240,  1740,   380;
  360, 43170, 81000, 70740, 31680, 6390;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..55 (rows 1..10)
Index entries for sequences related to monoids

Row sums give A058155.
Column 1: A034382.
Main diagonal: A055512.
Cf. A058142 (isomorphism classes), A058157, A058160.

T(n, k) = A058160(n, k)*n.

Terms a(30) and beyond from Andrew Howroyd, Feb 15 2022

A058160 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents and a fixed identity.

Original entry on oeis.org

1, 1, 1, 1, 6, 2, 4, 45, 36, 9, 6, 408, 648, 348, 76, 60, 7195, 13500, 11790, 5280, 1065, 120, 200016, 369930, 425280, 293640, 118890, 22566, 1920, 13811077, 14716716, 18035745, 16347660, 9953265, 3750516, 674611, 7560, 3405271616, 1024787568, 971651856, 999666920, 794967600, 445001424, 157962168, 27019896

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
   1;
   1,    1;,
   1,    6,     2;
   4,   45,    36,     9;
   6,  408,   648,   348,   76;
  60, 7195, 13500, 11790, 5280, 1065;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..55 (rows 1..10)
Index entries for sequences related to monoids

Row sums give A058156.
Column 1: A058162.
Main diagonal A058164.
Cf. A058142 (isomorphism classes), A058158, A058159.

T(n, k) = A058159(n, k)/n.

Terms a(30) and beyond from Andrew Howroyd, Feb 15 2022

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

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

A058143 Number of commutative monoids of order n with 2 idempotents.

Original entry on oeis.org

1, 3, 9, 26, 98, 455, 4018, 101910, 10054303

Offset: 2

Christian G. Bower, Nov 14 2000

Index entries for sequences related to monoids

Column 2 of A058142.

a(8)-a(10) from Andrew Howroyd, Feb 15 2022

cp's OEIS Frontend

A006966 Number of lattices on n unlabeled nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A058159 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A058160 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents and a fixed identity.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Extensions

A058143 Number of commutative monoids of order n with 2 idempotents.

Views

Author

Keywords

Links

Crossrefs

Extensions