A006982 -id:A006982 - 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

A006981 a(n) is the number of unlabeled modular lattices on n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 16, 34, 72, 157, 343, 766, 1718, 3899, 8898, 20475, 47321, 110024, 256791, 601991, 1415768, 3340847, 7904700, 18752943, 44588803, 106247120, 253644319, 606603025, 1453029516, 3485707007, 8373273835, 20139498217, 48496079939, 116905715114, 282098869730

Offset: 0

N. J. A. Sloane

nonn

			From _Jukka Kohonen_, Mar 06 2021: (Start)
a(5)=4: These are the four lattices.
  o     o       o        o
  |     |      / \      /|\
  o     o     o   o    o o o
  |    / \     \ /      \|/
  o   o   o     o        o
  |    \ /      |
  o     o       o
  |
  o
(End)

P. D. Lincoln, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jukka Kohonen, Table of n, a(n) for n = 0..35
R. Belohlavek and V. Vychodil, Residuated lattices of size <=12, Order 27 (2010) 147-161, Table 6.
D. J. Greenhoe, MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing, 2014.
P. Jipsen and N. Lawless, Generating all modular lattices of a given size, arXiv:1309.5036 [math.CO], 2013-2014.
J. Kohonen, Generating modular lattices up to 30 elements, arXiv:1708.03750 [math.CO] preprint (2017).
J. Kohonen, Cartesian lattice counting by the vertical 2-sum, arXiv:2007.03232 [math.CO] preprint (2020).
J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29. (Annotated scanned copy)

Cf. A006966 (lattices), A006982 (distributive), A342132 (modular vertically indecomposable).

More terms from Nathan Lawless, Sep 15 2013
Corrected a(24) and added a(25)-a(30) by Jukka Kohonen, Aug 15 2017
a(31)-a(32) from Jukka Kohonen, Sep 23 2018
Name clarified by Jukka Kohonen, Sep 23 2018
a(33) from Jukka Kohonen, Sep 26 2018
a(34)-a(35) from Jukka Kohonen, Mar 06 2021

A343161 Number of planar distributive lattices with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 14, 24, 42, 72, 127, 221, 390, 684, 1207, 2125, 3753, 6620, 11698, 20659, 36518, 64533, 114099, 201707, 356683, 630693, 1115370, 1972469, 3488489, 6169656, 10912003, 19299555, 34135099, 60374747, 106786342, 188875933, 334072759, 590889162, 1045136443

Offset: 1

N. J. A. Sloane, Apr 18 2021, following a suggestion from Allan C. Wechsler

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Peter Jipsen, Planar distributive lattices up to size 11, March 2014.
Peter Jipsen, Planar distributive lattices up to size 15, March 2014.
Peter Jipsen, Planar vertically indecomposable distributive lattices up to size 22, March 2014.

Cf. A006966, A006982, A345734.

V=concat(digits(1101010214296),[21,18,48,50,114,135,277,358,681]); P=List(1); for(n=2,#V,listput(P,V[2..n]*Colrev(P))); A343161=Vec(P) \\ M. F. Hasler, Jun 22 2021, using V[1..22] & formula from Bianca Newell

\\ Needs S, V defined in A345734.
seq(n)={Vec(x/(1 - x - Ser((S(n)+V(n))/2)))} \\ Andrew Howroyd, Jan 24 2023

v=[1,1,1,0,1,0,1,0,2,1,4,2,9,6,21,18,48,50,114,135,277,358,681]
p=[1,1,1]
for n in range(3,23):
    p=p+[sum(v[k]*p[n-k+1] for k in range(2,n+1))]
p # Bianca Newell, Jun 22 2021

a(n) = Sum_{k=2..n} V(k)*a(n-k+1), where V(k) is the number of planar vertically indecomposable distributive lattices of size k. - Bianca Newell, Jun 22 2021

G.f.: x/(2 - B(x)/x) where B(x) is the g.f of A345734. - Andrew Howroyd, Jan 24 2023

a(16)-a(22), computed with Python code, from Bianca Newell, Jun 22 2021

Terms a(23) and beyond from Andrew Howroyd, Jan 24 2023

A072361 Number of vertically indecomposable distributive lattices on n nodes.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 3, 1, 6, 2, 16, 8, 42, 28, 112, 93, 311, 295, 869, 939, 2454, 2931, 7032, 9036, 20301, 27701, 58929, 84413, 172104, 255919, 504637, 773511, 1484392, 2331180, 4378773, 7009288, 12944347, 21039961, 38328890, 63067623, 113651785, 188831922, 337361112, 564890985, 1002268019, 1688673026, 2979703035, 5045200597

Offset: 1

N. J. A. Sloane, Aug 01 2002

nonn

M. Erné, J. Heitzig and J. Reinhold, On the number of distributive lattices, Electronic Journal of Combinatorics, 9 (2002), #R24.

Cf. A006982, A072407.

Word "distributive" added to the Name by Peter Jipsen, Mar 05 2013

A072407 Canonically 2-indecomposable posets with n antichains.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 2, 1, 1, 0, 7, 5, 16, 14, 40, 41, 120, 131, 321, 402, 901, 1210, 2590, 3621, 7371, 10841, 21178, 32222, 61273, 95408, 177384, 282405, 515174, 833295, 1500030, 2455337, 4372535, 7229231, 12761691, 21260746, 37286778, 62483221, 109014426, 183542099, 318906720, 538889399, 933361886, 1581666042

Offset: 1

N. J. A. Sloane, Aug 01 2002

nonn

M. Erné, J. Heitzig and J. Reinhold, On the number of distributive lattices, Electronic Journal of Combinatorics, 9 (2002), #R24.

Cf. A006982, A072361.

A377408 Number of unlabeled semidistributive lattices with n elements.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 22, 60, 174, 534, 1720, 5767, 20013, 71546

Offset: 1

Ludovic Schwob, Oct 27 2024

The smallest semidistributive lattice that is not congruence-uniform has 14 elements.

H. Mühle, On the lattice property of shard orders, arXiv:1708.02104 [math.CO], 2017.

Cf. A006982 (distributive lattices), A292790 (congruence-uniform lattices).

A186808 Numbers n such that there are a prime number of unlabeled distributive lattices with n elements.

Original entry on oeis.org

4, 5, 6, 10, 12, 13, 18, 21, 23, 26

Offset: 1

Jonathan Vos Post, Feb 26 2011

nonn

A lattice which satisfies the identities:
(x^y)V(x^z) = x^(yVz);
(xVy)^(xVz) = xV(y^z)
is said to be distributive.

			a(10) = 26 because there are 711811 unlabeled distributive lattices with 26 elements, and 711811 is a prime number.

Gratzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, pp. 35-36, 1971.

Cf. A000040, A006982

{k: A006982(k) is in A000040}.

A377409 Number of unlabeled join-semidistributive lattices with n elements.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 23, 65, 197, 636, 2171, 7756, 28822, 110805

Offset: 1

Ludovic Schwob, Oct 27 2024

Equivalently, a(n) is the number of meet-semidistributive lattices with n elements.

The smallest join-semidistributive lattice that is not meet-semidistributive has 7 elements.

Cf. A377408 (semidistributive lattices), A006982 (distributive lattices).

A382829 Number of distinct rank vectors of distributive lattices of height n.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 197, 864, 4325, 24922

Offset: 0

Ludovic Schwob, Apr 06 2025

Distributive lattices are ranked posets, and we define the rank vector of a ranked poset P as the vector whose k-th coordinate (starting at k = 0) is the number of elements of rank k in P.

By Birkhoff's representation theorem, elements of a finite distributive lattice L are in bijection with lower sets of the poset of join-irreducible elements of L, an element of rank k corresponding to a lower of set size k.

			The rank vectors corresponding to a(4) = 15 are:
  (1, 1, 1, 1, 1),   (1, 1, 1, 2, 1),   (1, 1, 2, 1, 1),
  (1, 1, 2, 2, 1),   (1, 1, 3, 3, 1),   (1, 2, 1, 1, 1),
  (1, 2, 1, 2, 1),   (1, 2, 2, 1, 1),   (1, 2, 2, 2, 1),
  (1, 2, 3, 2, 1),   (1, 2, 3, 3, 1),   (1, 3, 3, 1, 1),
  (1, 3, 3, 2, 1),   (1, 3, 4, 3, 1),   (1, 4, 6, 4, 1).
Two non-isomorphic distributive lattices have for rank vector (1, 2, 2, 2, 1).

Cf. A000112, A006982.

cp's OEIS Frontend

A006966 Number of lattices on n unlabeled nodes.

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A006981 a(n) is the number of unlabeled modular lattices on n nodes.

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A343161 Number of planar distributive lattices with n nodes.

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A072361 Number of vertically indecomposable distributive lattices on n nodes.

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A072407 Canonically 2-indecomposable posets with n antichains.

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A377408 Number of unlabeled semidistributive lattices with n elements.

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A186808 Numbers n such that there are a prime number of unlabeled distributive lattices with n elements.

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A377409 Number of unlabeled join-semidistributive lattices with n elements.

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A382829 Number of distinct rank vectors of distributive lattices of height n.

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