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
KEYWORD
nonn,tabl
AUTHOR
Jukka Kohonen, Mar 01 2018
STATUS
approved