A006966 -id:A006966 - cp's OEIS frontend

A229202 Number of semimodular lattices on n nodes.

1, 1, 1, 1, 2, 4, 8, 17, 38, 88, 212, 530, 1376, 3693, 10232, 29231, 85906, 259291, 802308, 2540635, 8220218, 27134483, 91258141, 312324027, 1086545705, 3838581926

Offset: 0

Nathan Lawless, Sep 15 2013

P. Jipsen and N. Lawless, Generating all modular lattices of a given size
J. Kohonen, Generating modular lattices up to 30 elements, arXiv:1708.03750 [math.CO] preprint (2017).

Cf. A006966 (number of lattices), A006981 (number of modular lattices).

a(23)-a(25) from Kohonen (2017), by Jukka Kohonen, Aug 15 2017

A030268 Number of nonisomorphic connected partial lattices.

Original entry on oeis.org

1, 1, 1, 3, 9, 35, 153, 791, 4597, 29988, 215804, 1697291, 14457059, 132392971, 1295346365, 13468653637, 148142236784, 1716782858995, 20889118889021

Offset: 0

Christian G. Bower, revised Dec 28 2000

A partial lattice is a poset where every pair of points has a unique least upper (greatest lower) bound or has no upper (lower) bound.

N. J. A. Sloane, Transforms

A006966 = Cases[Import["https://oeis.org/A006966/b006966.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
Join[{1}, EulerInvTransform[Drop[A006966, 3]]] (* Jean-François Alcover, May 10 2019, updated Mar 17 2020 *)

Inverse Euler transform of A006966(n-2) (lattices).

a(17) (from A006966) from Jean-François Alcover, May 10 2019

a(18) (using A006966) from Alois P. Heinz, May 10 2019

A278691 Number of graded lattices on n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 22, 60, 176, 565, 1980, 7528, 30843, 135248, 630004, 3097780, 15991395, 86267557, 484446620, 2822677523, 17017165987

Offset: 1

Jukka Kohonen, Nov 26 2016

A finite lattice is graded if, for any element, all paths from the bottom to that element have the same length.

J. Heitzig and J. Reinhold, Counting finite lattices, Algebra Universalis, 48 (2002), 43-53.
J. Kohonen, Generating modular lattices up to 30 elements, arXiv:1708.03750 [math.CO] preprint (2017).
M. Malandro, The unlabeled lattices on <=15 nodes, (listing of lattices; graded lattices are a subset of these).

Cf. A006966 (lattices), A229202 (semimodular lattices).

a(16)-a(21) from Kohonen (2017), by Jukka Kohonen, Aug 15 2017

A373894 Number of self-dual lattices on n unlabeled nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 13, 36, 76, 232, 562, 1860, 5025

Offset: 0

Jukka Kohonen, Jun 21 2024

Lattices whose Hasse diagram looks the same if it is turned upside down.

			a(5)=3: These are the three lattices.
  o      o        o
  |     / \      /|\
  o    o   |    o o o
  |    |   o     \|/
  o    o   |      o
  |     \ /
  o      o
  |
  o

Volker Gebhardt and Stephen Tawn, Catalogue of unlabelled lattices on up to 16 elements, Western Sydney University (2018).

Cf. A006966 (lattices), A133983 (self-dual posets).

sum(L.is_lattice() and L.is_self_dual() for L in Posets(n))

A030269 Number of nonisomorphic disconnected partial lattices.

Original entry on oeis.org

0, 0, 1, 2, 6, 18, 69, 287, 1397, 7634, 46972, 321014, 2416305, 19840547, 176267022, 1681915809, 17127587977, 185127766583

Offset: 0

Christian G. Bower, revised Dec 28 2000

A partial lattice is a poset where every pair of points has a unique least upper (greatest lower) bound or has no upper (lower) bound.

a(n) = A006966(n-2) - A030268(n).

A058801 Number of connected vertically indecomposable partial lattices on n unlabeled nodes.

Original entry on oeis.org

1, 2, 6, 25, 116, 625, 3757, 25140, 184511, 1473861, 12711339, 117598686, 1160399052, 12152333659, 134487937252, 1566878426731, 19154490559458

Offset: 2

Christian G. Bower, Dec 28 2000

A partial lattice is a poset where every pair of points has a unique least upper (greatest lower) bound or has no upper (lower) bound.

N. J. A. Sloane, Transforms

Cf. A006966.

A058800 = Cases[Import["https://oeis.org/A058800/b058800.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
EulerInvTransform[Drop[A058800, 3]] // Rest (* Jean-François Alcover, May 10 2019, updated Mar 17 2020 *)

Inverse EULER transform of A058800(n+2).

a(17)-a(18) (computed from A058800) from Jean-François Alcover, May 10 2019

A058802 Vertically decomposable lattices on n unlabeled nodes.

Original entry on oeis.org

1, 1, 3, 8, 26, 96, 414, 2040, 11432, 72022, 503973, 3875329, 32429747, 292872455, 2834089224, 29209213572, 318979706486

Offset: 3

Christian G. Bower, Dec 28 2000

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.

Cf. A006966, A058800.

a(n) = A006966(n) - A058800(n).

A159483 Number of tolerance simple lattices of order n.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 4, 14, 71, 389

Offset: 1

Tim Boykett (tim(AT)timesup.org), Apr 14 2009

nonn

Relevant for order polynomial completeness of lattices and tame congruence theory.

			The unique lattices of order 1 and 2 are tolerance simple. The lattices of order 3 and 4 are not. M_3 is of order 5 and is tolerance simple. M_4 is of order 6 and is tolerance simple. Then it gets complicated.

K.Kaarli and A.F.Pixley, Polynomial Completeness in Algebraic Systems, Chapman & Hall/CRC, 2001

Number of lattices is A006966

A173488 Partial sums of A055512.

Original entry on oeis.org

1, 2, 4, 10, 46, 426, 6816, 164778, 5561666, 248740730, 14187451940, 1002045820690, 85615117761142, 8682866612715706, 1029036311254555560, 140656568448867136650, 21929110364021381812410, 3862525357012048643891882, 762298016068721625860646524

Offset: 0

Jonathan Vos Post, Feb 19 2010

Partial sums of the number of lattices with n labeled elements. After a(0) = 1, always even, hence the only prime in the partial sum is 2. The subsequence of semiprimes begins 4, 10, 46.

Cf. A055512, A006966, A001035, Main diagonal of A058159.

Cases[Import["https://oeis.org/A055512/b055512.txt", "Table"], {, }][[All, 2]] // Accumulate (* Jean-François Alcover, Jan 02 2020 *)

a(n) = Sum_{i=0..n} A055512(i).

a(17)-a(18) from Jean-François Alcover, Jan 02 2020

A271077 Number of pseudocomplemented lattices on n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 10, 29, 99, 391, 1775, 9214

Offset: 0

Jori Mäntysalo, Mar 30 2016

R. Belohlavek, V. Vychodil, Residuated lattices of size <=12, Order 27 (2010) 147-161 doi:10.1007/s11083-010-9143-7, Table 6.
Wikipedia, pseudocomplement.

Cf. A006966.

for i in range(0, 12):
    n = 0
    for P in Posets(i):
        if P.is_lattice():
            L = LatticePoset(P)
            if L.is_pseudocomplemented():
                n += 1
    print(n)

cp's OEIS Frontend

A229202 Number of semimodular lattices on n nodes.

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A030268 Number of nonisomorphic connected partial lattices.

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A278691 Number of graded lattices on n nodes.

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A373894 Number of self-dual lattices on n unlabeled nodes.

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A030269 Number of nonisomorphic disconnected partial lattices.

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A058801 Number of connected vertically indecomposable partial lattices on n unlabeled nodes.

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A058802 Vertically decomposable lattices on n unlabeled nodes.

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A159483 Number of tolerance simple lattices of order n.

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A173488 Partial sums of A055512.

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A271077 Number of pseudocomplemented lattices on n nodes.

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