OFFSET
1,3
COMMENTS
Colors are not being permuted, i.e., Power Group Enumeration does not apply here.
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequences A294684, A294685, A294686, A294687.
FORMULA
T(n,k) = (Q!/(n*k))*(Sum_{d|n} Sum_{f|k} phi(d) phi(f) S(gcd(d,f)*(n/d)*(k/f), Q)) with Q=2 and S(n,k) Stirling numbers of the second kind.
T(n,k) = A184271(n,k) - 2. - Andrew Howroyd, Oct 05 2024
EXAMPLE
Triangle begins:
0;
1, 5;
2, 12, 62;
4, 38, 350, 4154;
6, 106, 2190, 52486, 1342206;
12, 360, 14622, 699598, 35792566, 1908897150;
18, 1180, 99878, 9587578, 981706830, 104715443850, 11488774559742;
...
For the 2 X 2 and two colors we find
+---+ +---+ +---+ +---+ +---+
|X| | | |X| |X| | |X|X| |X| |
+-+-+ +-+-+ +-+-+ +-+-+ +-+-+
| | | |X|X| | |X| | | | |X| |
+-+-+ +-+-+ +-+-+ +-+-+ +-+-+
MATHEMATICA
With[{Q = 2}, Table[(Q!/n/k) Sum[Sum[EulerPhi[d] EulerPhi[f] StirlingS2[GCD[d, f] (n/d) (k/f), Q], {f, Divisors@ k}], {d, Divisors@ n}], {n, 8}, {k, n}]] // Flatten (* Michael De Vlieger, Nov 08 2017 *)
PROG
(PARI) T(n, m) = {2*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d, e), 2, 2) ))/(n*m)} \\ Andrew Howroyd, Oct 05 2024
CROSSREFS
KEYWORD
AUTHOR
Marko Riedel, Nov 06 2017
STATUS
approved