A006981 -id:A006981 - 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

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.

A229202 Number of semimodular lattices on n nodes.

Original entry on oeis.org

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

A342132 Number of unlabeled vertically indecomposable modular lattices on n nodes.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 3, 7, 12, 28, 54, 127, 266, 614, 1356, 3134, 7091, 16482, 37929, 88622, 206295, 484445, 1136897, 2682451, 6333249, 15005945, 35595805, 84649515, 201560350, 480845007, 1148537092, 2747477575, 6579923491, 15777658535, 37871501929

Offset: 1

Jukka Kohonen, Mar 01 2021

nonn

A lattice is vertically decomposable if it has an element that is comparable to all elements and is neither the bottom nor the top element. Otherwise the lattice is vertically indecomposable.

			a(7)=3: These are the three lattices.
      o        o         __o__
     / \      /|\       / /|\ \
    o   o    o o o     o o o o o
   /|\ /    / \|/       \_\|/_/
  o o o    o   o           o
   \|/      \ /
    o        o

P. Jipsen and N. Lawless, Generating all finite modular lattices of a given size, Algebra universalis, 74 (2015), 253-264.
J. Kohonen, Generating modular lattices of up to 30 elements, Order, 36 (2019), 423-435.
J. Kohonen, Cartesian lattice counting by the vertical 2-sum, arXiv:2007.03232 [math.CO] preprint (2020).

Cf. A006981 (modular lattices, including vertically decomposable).

A368461 a(n) is the number of unlabeled planar modular lattices on n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 33, 70, 151, 329, 723, 1601, 3569, 8000, 18015, 40723, 92351, 209997, 478598, 1092856, 2499567, 5724970, 13128115, 30135636, 69238343, 159202607, 366308948, 843338278, 1942591448, 4476714720, 10320774953, 23802355725, 54911686727

Offset: 1

Jukka Kohonen, Dec 25 2023

nonn

Jukka Kohonen, Modular racks (code to compute the numbers).

Cf. A006981 (modular lattices), A343161 (planar distributive lattices).

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A006966 Number of lattices on n unlabeled nodes.

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A006982 Number of unlabeled distributive lattices on n nodes.

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A229202 Number of semimodular lattices on n nodes.

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A342132 Number of unlabeled vertically indecomposable modular lattices on n nodes.

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A368461 a(n) is the number of unlabeled planar modular lattices on n nodes.

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