A055512 -id:A055512 - cp's OEIS frontend

A173488 Partial sums of A055512.

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

A006966 Number of lattices on n unlabeled nodes.

Original entry on oeis.org

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

A058164 Number of labeled lattices with a fixed bottom.

Original entry on oeis.org

1, 1, 2, 9, 76, 1065, 22566, 674611, 27019896, 1393871121, 89805306250, 7051089328371, 661327038073428, 72882388902988561, 9308502142507505406, 1361778362223282167235, 225917426273413368357616, 42135305039539420956486369, 8768279861975723002123310226

Offset: 1

Christian G. Bower, Nov 15 2000

Main diagonal of A058160.

Cf. A055512, A058165.

A055512 = Cases[Import["https://oeis.org/A055512/b055512.txt", "Table"], {, }][[All, 2]];
a[n_] := A055512[[n + 1]]/n;
a /@ Range[18] (* Jean-François Alcover, Jan 02 2020 *)

a(n) = A055512(n)/n = A058165(n)*(n-1).

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

a(19) from the data at A055512 added by Amiram Eldar, Jul 22 2025

A058165 Number of labeled lattices with a fixed bottom and top.

Original entry on oeis.org

1, 1, 3, 19, 213, 3761, 96373, 3377487, 154874569, 8980530625, 641008120761, 55110586506119, 5606337607922197, 664893010179107529, 90785224148218811149, 14119839142088335522351, 2478547355267024762146257

Offset: 2

Christian G. Bower, Nov 15 2000

Cf. A055512, A058164.

A055512 = Cases[Import["https://oeis.org/A055512/b055512.txt", "Table"], {, }][[All, 2]];
a[n_] := A055512[[n+1]]/(n(n-1));
a /@ Range[2, 18] (* Jean-François Alcover, Jan 02 2020 *)

a(n) = A055512(n)/(n*(n-1)) = A058164(n)/(n-1).

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

cp's OEIS Frontend

A173488 Partial sums of A055512.

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A006966 Number of lattices on n unlabeled nodes.

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

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A058164 Number of labeled lattices with a fixed bottom.

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A058165 Number of labeled lattices with a fixed bottom and top.

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