Jukka Kohonen - cp's OEIS frontend

A373922 Number of lattices on n unlabeled nodes, up to duality.

1, 1, 1, 1, 2, 4, 11, 33, 129, 577, 3113, 19092, 132318, 1011665

Offset: 0

Jukka Kohonen, Jun 30 2024

Number of nonisomorphic lattices on n nodes, when from each pair of dual lattices only one is counted.

			a(5)=4: These are the four lattices. The dual of the last one is not counted.
  o      o        o       o
  |     / \      /|\      |
  o    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).

a(n) = (A006966(n) + A373894(n)) / 2.

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))

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).

A347025 Maximum size of a family of nonempty sets on n elements such that no set is a union of some others.

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 24

Offset: 0

Jukka Kohonen, Sep 29 2021

The upper bounds of Loo (table on pp. 11-13; formula below) may be improved given the term a(5). Specifically, using h = 1 and a(5) in Loo's upper bound formula yields a(6) <= 27 (versus the published 30). The lower and upper bounds may be used to distinguish this sequence from others in the OEIS. - Michael S. Branicky, Mar 16 2022
a(7) >= 44, a(8) >= 79, a(9) >= 144, a(10) >= 270; see the Apr 05 2022 entry in the Formula section. - Jon E. Schoenfield, Apr 04 2022
a(7) <= 45. - Jinyuan Wang, Apr 23 2022

			a(4) = 7: an example of such a family is {{1},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}.

Michael S. Branicky, Python program
Andy Loo, Union-Free Families of Subsets, arXiv:1511.00170 [math.CO], 2015.
Augusto Perez, Maximum size of antichain-like collection, Math.SE question (2021).

from itertools import combinations
def anysetunion(family):
    for s in family:
        allrest = 0
        for r in family:
            if r != s and r&s == r:
                allrest |= r
                if allrest == s:
                    return True
    return False
def a(n):
    if n < 2: return n
    m = 2
    while True:
        allfailed = True
        for family in combinations(range(1, 2**n), m):
            unionfound = anysetunion(family)
            allfailed &= unionfound
            if not unionfound: break
        if allfailed: return m - 1
        m += 1
print([a(n) for n in range(5)]) # Michael S. Branicky, Nov 09 2021

From Michael S. Branicky, Mar 16 2022: (Start)
Bounds from Loo (p. 10):
  a(n) >= binomial(n, ceiling(n/2)),
  a(n) >= max_{h=1..n-1} a(h) + a(n-h) + 1,
  a(n) <= min_{h=1..n-1} a(h) + 2^h*a(n-h). (End)
For n > 2, a(n) >= max_{m=3..n} 2*floor(m/3) + binomial(m,3) + [n < 6] + Sum_{j=m..n-1} binomial(j,m-3) where [n < 6] is an Iverson bracket. - Jon E. Schoenfield, Apr 05 2022

a(6) from Jinyuan Wang, Apr 19 2022

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).

A306608 Table read by antidiagonals: T(x,y) is the minimum size of a planar additive basis for the rectangle [0,x]*[0,y], for x,y >= 0.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 4, 4, 3, 3, 5, 4, 5, 3, 4, 5, 6, 6, 5, 4, 4, 6, 6, 7, 6, 6, 4, 4, 6, 7, 8, 8, 7, 6, 4, 4, 7, 8, 9, 8, 9, 8, 7, 4, 5, 7, 8, 9, 10, 10, 9, 8, 7, 5, 5, 8, 8, 10, 10, 11, 10, 10, 8, 8, 5, 5, 8, 10, 11, 11, 12, 12, 11, 11, 10, 8, 5

Offset: 0

Jukka Kohonen, Feb 28 2019

A planar additive basis is a set of points with nonnegative integer coordinates such that their pairwise sums cover a given rectangle of points with integer coordinates. Pairwise sums of a point with itself are included.

T(x,y) = T(y,x).

			The table starts:
  1, 2, 2, 3, 3, 4, 4, ...
  2, 3, 4, 5, 5, 6, ...
  2, 4, 4, 6, 6, ...
  3, 5, 6, 7, ...
  3, 5, 6, ...
  4, 6, ...
  4, ...
  ...
T(6,3)=9: The rectangle [0,6]*[0,3] has the following minimum basis of 9 elements, with elements marked as "*", and empty locations as "-".
  3  *------
  2  ---*---
  1  **-*---
  0  ***--*-
     0123456

J. Kohonen, V. Koivunen and R. Rajamäki, Planar additive bases for rectangles, Journal of Integer Sequences, 21 (2018), Article 18.9.8. [see Table 2]

Main diagonal is A295771.

A322600 a(n) is the number of unlabeled rank-3 graded lattices with 5 coatoms and n atoms.

Original entry on oeis.org

1, 5, 20, 68, 190, 441, 907, 1690, 2916, 4734, 7310, 10836, 15528, 21619, 29365, 39045, 50961, 65434, 82809, 103453, 127751, 156117, 188980, 226794, 270037, 319204, 374813, 437409, 507553, 585831, 672847, 769233, 875637, 992735, 1121218, 1261802

Offset: 1

Jukka Kohonen, Dec 19 2018

Jukka Kohonen, Table of n, a(n) for n = 1..1000
J. Kohonen, Counting graded lattices of rank three that have few coatoms, arXiv:1804.03679 [math.CO] preprint (2018).

Fifth row of A300260.

Previous rows are A322598, A322599.

For n>=3: a(n) = (175/192)n^4 - (3079/480)n^3 + (11771/480)n^2
  - [7268/160, 7273/160]n
  + [33600, 34019, 34072, 33627, 33152, 34915, 33624, 33947, 33472, 33507,
  34520, 34459, 32832, 33827, 34072, 34395, 33344, 34147, 33432, 33947,
  34240, 33699, 33752, 34267, 32832, 34595, 34264, 33627, 33152, 34147,
  34200, 34139, 33472, 33507, 33752, 35035, 33024, 33827, 34072, 33627,
  33920, 34339, 33432, 33947, 33472, 34275, 33944, 34267, 32832, 33827,
  34840, 33819, 33152, 34147, 33432, 34715, 33664, 33507, 33752, 34267] / 960.
  The value of the first bracket depends on whether n is even or odd. The value of the second bracket depends on whether (n mod 60) is 0, 1, 2, ..., 59.
Conjectures from Colin Barker, Dec 20 2018: (Start)
G.f.: x*(1 + 4*x + 14*x^2 + 43*x^3 + 102*x^4 + 184*x^5 + 282*x^6 + 368*x^7 + 411*x^8 + 400*x^9 + 333*x^10 + 237*x^11 + 142*x^12 + 70*x^13 + 26*x^14 + 7*x^15 + x^16) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-6) - a(n-7) + a(n-8) + a(n-9) + a(n-10) - a(n-13) - a(n-14) + a(n-15) for n>15.
(End)

A322599 a(n) is the number of unlabeled rank-3 graded lattices with 4 coatoms and n atoms.

Original entry on oeis.org

1, 4, 13, 34, 68, 121, 197, 299, 432, 600, 806, 1055, 1352, 1698, 2100, 2561, 3085, 3675, 4338, 5074, 5891, 6790, 7777, 8854, 10029, 11300, 12677, 14160, 15756, 17465, 19297, 21249, 23332, 25544, 27894, 30381, 33016, 35794, 38728, 41815, 45065

Offset: 1

Jukka Kohonen, Dec 19 2018

			a(2)=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
   \_\ /_/|      \|/ \|      \|/  |     |/   \|
      o   o       o   o       o   o     o     o
       \ /         \ /         \ /       \_ _/
        o           o           o          o

Jukka Kohonen, Table of n, a(n) for n = 1..1000
J. Kohonen, Counting graded lattices of rank three that have few coatoms, arXiv:1804.03679 [math.CO] preprint (2018).

Fourth row of A300260.

Adjacent rows are A322598, A322600.

a(n) = (97/144)n^3 - (5/6)n^2 + [44/48, 47/48]n + [0, 13, 8, -45, 40, -19, 0, -5, 8, -27, 40, -37]/72. The value of the first bracket depends on whether n is even or odd. The value of the second bracket depends on whether (n mod 12) is 0, 1, 2, ..., 11.
Conjectures from Colin Barker, Dec 19 2018: (Start)
G.f.: x*(1 + 3*x + 8*x^2 + 17*x^3 + 21*x^4 + 21*x^5 + 16*x^6 + 7*x^7 + 3*x^8) / ((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n>10.
(End)

A322598 a(n) is the number of unlabeled rank-3 graded lattices with 3 coatoms and n atoms.

Original entry on oeis.org

1, 3, 8, 13, 20, 29, 39, 50, 64, 78, 94, 112, 131, 151, 174, 197, 222, 249, 277, 306, 338, 370, 404, 440, 477, 515, 556, 597, 640, 685, 731, 778, 828, 878, 930, 984, 1039, 1095, 1154, 1213, 1274, 1337, 1401, 1466, 1534, 1602, 1672, 1744, 1817

Offset: 1

Jukka Kohonen, Dec 19 2018

Also number of bicolored graphs, with 3 vertices in the first color class and n in the second, with no isolated vertices, and where any two vertices in one class have at most one common neighbor.

			a(2)=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

Jukka Kohonen, Table of n, a(n) for n = 1..1000
J. Kohonen, Counting graded lattices of rank three that have few coatoms, arXiv:1804.03679 [math.CO] preprint (2018).
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Third row of A300260.

Next rows are A322599, A322600.

List([1..50],n->Int((3/4)*n^2+(1/3)*n+1/4)); # Muniru A Asiru, Dec 20 2018

seq(floor(3/4*n^2+n/3+1/4),n=1..100); # Robert Israel, Dec 19 2018

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 3, 8, 13, 20, 29}, 50] (* Jean-François Alcover, Dec 29 2018 *)

Vec(x*(1 + 2*x + 4*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)) + O(x^50)) \\ Colin Barker, Dec 19 2018

a(n) = floor( (3/4)n^2 + (1/3)n + 1/4 ).
From Colin Barker, Dec 19 2018: (Start)
G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>6.
(End)
From Robert Israel, Dec 19 2018: (Start)
a(6*m) = 27*m^2+2*m.
a(6*m+1) = 27*m^2+11*m+1.
a(6*m+2) = 27*m^2+20*m+3.
a(6*m+3) = 27*m^2+29*m+8.
a(6*m+4) = 27*m^2+38*m+13.
a(6*m+5) = 27*m^2+47*m+20.
These imply the conjectured G.f. and recursion.(End)

A300221 a(n) is the number of unlabeled, graded rank-3 lattices with n elements.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 8, 18, 38, 88, 210, 528, 1396, 3946, 11896, 38644, 135790, 518645, 2160112, 9832013, 48945468, 266458643

Offset: 1

Jukka Kohonen, Mar 01 2018

A graded lattice has rank 3 if its maximal chains have length 3.
They can be enumerated with a program such as that by Kohonen (2017).
Also called "two level lattices": apart from top and bottom, they have just coatoms and atoms. (Kleitman and Winston 1980)
Asymptotic upper bound: a(n) < b^(n^(3/2) + o(n^(3/2))), where b is about 1.699408. (Kleitman and Winston 1980)

			a(4)=1: The only possibility is the chain of length 3 (with 4 elements).
a(6)=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
   \|/     \ /    \ /      |
    o       o      o       o

D. J. Kleitman and K. J. Winston, The asymptotic number of lattices, Ann. Discrete Math. 6 (1980), 243-249.
J. Kohonen, Generating modular lattices up to 30 elements, arXiv:1708.03750 [math.CO] preprint (2017).

Cf. A278691 (unlabeled graded lattices).

a(n) = Sum_{k=1..n-3} A300260(n-2-k, k).

a(22) from Jukka Kohonen, Mar 03 2018

cp's OEIS Frontend

User: Jukka Kohonen

A373922 Number of lattices on n unlabeled nodes, up to duality.

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

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Sage

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

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A347025 Maximum size of a family of nonempty sets on n elements such that no set is a union of some others.

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Python

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

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A306608 Table read by antidiagonals: T(x,y) is the minimum size of a planar additive basis for the rectangle [0,x]*[0,y], for x,y >= 0.

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A322600 a(n) is the number of unlabeled rank-3 graded lattices with 5 coatoms and n atoms.

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A322599 a(n) is the number of unlabeled rank-3 graded lattices with 4 coatoms and n atoms.

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A322598 a(n) is the number of unlabeled rank-3 graded lattices with 3 coatoms and n atoms.

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