A006966 -id:A006966 - cp's OEIS frontend

A099085 A bisection of A006966.

Original entry on oeis.org

1, 1, 2, 15, 222, 5994, 262776, 16873364, 1471613387, 165269824761

Offset: 0

N. J. A. Sloane, Nov 15 2004

nonn

A099086 A bisection of A006966.

Original entry on oeis.org

1, 1, 5, 53, 1078, 37622, 2018305, 152233518, 15150569446

Offset: 0

N. J. A. Sloane, Nov 15 2004

nonn

A006982 Number of unlabeled distributive lattices on n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 8, 15, 26, 47, 82, 151, 269, 494, 891, 1639, 2978, 5483, 10006, 18428, 33749, 62162, 114083, 210189, 386292, 711811, 1309475, 2413144, 4442221, 8186962, 15077454, 27789108, 51193086, 94357143, 173859936, 320462062, 590555664, 1088548290, 2006193418, 3697997558, 6815841849, 12563729268, 23157428823, 42686759863, 78682454720, 145038561665, 267348052028, 492815778109, 908414736485

Offset: 0

N. J. A. Sloane

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..60
R. Belohlavek and V. Vychodil, Residuated lattices of size <=12, Order 27 (2010) 147-161, Table 6; DOI:10.1007/s11083-010-9143-7; Extended version.
Aaron Chan, Erik Darpö, Osamu Iyama, and René Marczinzik, Periodic trivial extension algebras and fractionally Calabi-Yau algebras, arXiv:2012.11927 [math.RT], 2020.
M. Erné, J. Heitzig and J. Reinhold, On the number of distributive lattices, Electronic Journal of Combinatorics, 9 (2002), #R24.
D. J. Greenhoe, MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing, 2014.
J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order 17 (2000) no. 4, 333-341.
J. Heitzig and J. Reinhold, Counting finite lattices, preprint no. 298, Institut für Mathematik, Universität Hanover, Germany, 1999.
J. Heitzig and J. Reinhold, Counting finite lattices, Algebra Universalis, 48 (2002), 43-53.
Institut f. Mathematik, Univ. Hanover, Erne/Heitzig/Reinhold papers
P. Jipsen, Planar distributive lattices up to size 15 (illustration of a(1..15)), personal web page, March 2014.
P. Jipsen and N. Lawless, Generating all finite modular lattices of a given size, 2013.
Jukka Kohonen, Cartesian lattice counting by the vertical 2-sum, Order (2021); see also on arXiv, arXiv:2007.03232 [math.CO], 2020.

Cf. A006981, A006966, A343161.

More terms from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Feb 02 2001. These were computed by the same algorithm that was used to enumerate the posets on 14 elements.

A354493 Number of quantales on n elements, up to isomorphism.

Original entry on oeis.org

1, 2, 12, 129, 1852, 33391, 729629, 19174600, 658343783

Offset: 1

Arman Shamsgovara, May 28 2022

A quantale is an algebraic structure (X,*,v) composed of a set X of elements, a semigroup operator "*" and a supremum operator "v" (in the sense of lattices) such that * distributes over v: x * (y v z) = (x * y) v (x * z) and (x v y) * z = (x * z) v (y * z) for all elements x,y,z in X. In addition the bottom element corresponding to v, denoted 0, must satisfy x * 0 = 0 * x = 0.

P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.
K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.
Arman Shamsgovara, A catalogue of every quantale of order up to 9 (abstract, to appear), LINZ2022, 39th Linz Seminar on Fuzzy Set Theory, Linz, Austria.
Arman Shamsgovara and P. Eklund, A Catalogue of Finite Quantales, GLIOC Notes, December 2019.

W. McCune, Prover9 and Mace4.
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.

Related algebraic structures: A027851, A006966.

assign(max_models,-1).
assign(domain_size,4).
formulas(assumptions).
% Comment: This will find all quantales on 4 elements, fixing
% 0 as the bottom and 3 as the top. Elements will be numbered
% 0-3. Results *must* be run through the companion program
% isofilter that is included with the downloads for mace4,
% otherwise the output will contain isomorphic duplicates!
% By changing the domain size, this file should be sufficient
% for up to 6 elements, but will crash for higher numbers.
(x*y)*z = x*(y*z).
(x v y) v z = x v (y v z).
x v y = y v x.
x v x = x.
x*(y v z) = (x*y) v (x*z).
(x v y)*z = (x*z) v (y*z).
0*x = 0.
x*0 = 0.
0 v x = x.
3 v x = 3.
end_of_list.
formulas(goals).
end_of_list.

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

A055512 Lattices with n labeled elements.

Original entry on oeis.org

1, 1, 2, 6, 36, 380, 6390, 157962, 5396888, 243179064, 13938711210, 987858368750, 84613071940452, 8597251494954564, 1020353444641839854, 139627532137612581090, 21788453795572514675760, 3840596246648027262079472, 758435490711709577216754642

Offset: 0

Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000

Sean A. Irvine, Table of n, a(n) for n = 0..19 (terms 0..18 from David Wasserman)
J. Heitzig and J. Reinhold, Counting finite lattices, preprint no. 298, Institut für Mathematik, Universität Hanover, Germany, 1999.
Sean A. Irvine, Java program (github).
J. Heitzig and J. Reinhold, Counting finite lattices, Algebra univers. 48, 43-53 (2002).
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.
Alan Veliz-Cuba and Reinhard Laubenbacher, Dynamics of semilattice networks with strongly connected dependency graph, Automatica (2019) Vol. 99, 167-174.
Index entries for "core" sequences

Cf. A006966, A001035. Main diagonal of A058159.

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

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

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

A058800 Vertically indecomposable lattices on n unlabeled nodes.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 7, 27, 126, 664, 3954, 26190, 190754, 1514332, 12998035, 119803771, 1178740932, 12316480222, 136060611189, 1582930919092, 19328253734491

Offset: 0

Christian G. Bower, Dec 28 2000

J. Heitzig and J. Reinhold, Counting finite lattices, Algebra Universalis, 48 (2002), 43-53.

V. Gebhardt and S. Tawn, Constructing unlabelled lattices, arXiv:1609.08255 [math.CO], 2016.
J. Heitzig and J. Reinhold, Counting finite lattices, preprint no. 298, Institut für Mathematik, Universität Hannover, Germany, 1999.
N. J. A. Sloane, Transforms

a(n+1) is Inverse INVERT transform of A006966(n+1).

A006966 = Cases[Import["https://oeis.org/A006966/b006966.txt", "Table"], {, }][[All, 2]];
nmax = Length[A006966] - 1;
B[x_] = Sum[A006966[[n + 1]] x^n, {n, 0, nmax}];
A[x_] = Sum[c[n] x^n, {n, 0, nmax}];
sol = CoefficientList[1 + A[x] - 1/(1 - B[x]) + O[x]^nmax, x] == 0 // Solve // First // Rest // Quiet;
a[n_] := If[n <= 2, 1, c[n - 2] /. sol];
a /@ Range[0, nmax] (* Jean-François Alcover, Dec 05 2019 *)

a(19) (computed by Jipsen and Lawless) and a(20) from Volker Gebhardt, Sep 28 2016

cp's OEIS Frontend

A099085 A bisection of A006966.

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A099086 A bisection of A006966.

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A006982 Number of unlabeled distributive lattices on n nodes.

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A354493 Number of quantales on n elements, up to isomorphism.

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Mace4

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

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A055512 Lattices with n labeled elements.

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

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A343161 Number of planar distributive lattices with n nodes.

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PARI

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A058800 Vertically indecomposable lattices on n unlabeled nodes.

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