A300260 -id:A300260 - cp's OEIS frontend

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

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)

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)

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)

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

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

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

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