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A300260 Table read by antidiagonals: T(n,k) is the number of unlabeled rank-3 graded lattices with n coatoms and k atoms (for n,k >= 1). 5
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 20, 34, 20, 6, 1, 1, 7, 29, 68, 68, 29, 7, 1, 1, 8, 39, 121, 190, 121, 39, 8, 1, 1, 9, 50, 197, 441, 441, 197, 50, 9, 1, 1, 10, 64, 299, 907, 1384, 907, 299, 64, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

T(n,k) = T(k,n), since taking the duals of the lattices swaps n and k.

Number of bicolored graphs, with n and k vertices in the color classes, with no isolated vertices, and where any two vertices in one class have at most one common neighbor. - Jukka Kohonen, Mar 08 2018

LINKS

Jukka Kohonen, Table of n, a(n) for n = 1..210

J. Kohonen, Counting graded lattices of rank three that have few coatoms, arXiv:1804.03679 [math.CO] preprint (2018).

FORMULA

T(2,k) = k. Proof: If the coatoms do not have a common atom, the k atoms can be divided between the two coatoms so that the smaller subset has 1..floor(k/2) atoms. If the coatoms have a common atom, the remaining k-1 can be divided so that the smaller subset has 0..floor((k-1)/2) atoms. In total this makes k possibilities. - Jukka Kohonen, Mar 03 2018

From Jukka Kohonen, Apr 20 2018 (Start)

T(3,k) = floor( (3/4)k^2 + (1/3)k + 1/4 )

T(4,k) = (97/144)k^3 - (5/6)k^2 + [44/48, 47/48]k + [0, 13, 8, -45, 40, -19, 0, -5, 8, -27, 40, -37]/72. The value of the first bracket depends on whether k is even or odd. The value of the second bracket depends on whether (k mod 12) is 0, 1, 2, ..., 11.

Formulas from (Kohonen 2018).

(End)

EXAMPLE

The table starts:

  1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...

  1,   2,   3,   4,   5,   6,   7,   8,   9, ...

  1,   3,   8,  13,  20,  29,  39,  50, ...

  1,   4,  13,  34,  68, 121, 197, ...

  1,   5,  20,  68, 190, 441, ...

  1,   6,  29, 121, 441, ...

  1,   7,  39, 197, ...

  1,   8,  50, ...

  1,   9, ...

  1, ...

  ...

PROG

(nauty) genbg -Z1 -d1 -u ${n} ${k}   # Jukka Kohonen, Mar 08 2018

CROSSREFS

Sum of the d-th antidiagonal is A300221(d+3).

Rows 3-5 are A322598, A322599, A322600.

Sequence in context: A135597 A169945 A073134 * A026692 A114202 A125806

Adjacent sequences:  A300257 A300258 A300259 * A300261 A300262 A300263

KEYWORD

nonn,tabl

AUTHOR

Jukka Kohonen, Mar 01 2018

STATUS

approved

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Last modified March 19 19:32 EDT 2019. Contains 321330 sequences. (Running on oeis4.)