OFFSET
1,2
COMMENTS
Two colorings are equivalent if there is a permutation of the colors that takes one to the other in addition to translational symmetries on the torus (Power Group Enumeration). The maximum number of colors is n * n.
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..24
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequence by PGE.
FORMULA
a(n) = Sum_{Q=1..n^2} (1/(n^2*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|n} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(n/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..Q} (exp(lz)-1)^j_l(sigma). The notation j_l(sigma) is from the Harary text and gives the number of cycles of length l in the permutation sigma. [[.]] is an Iverson bracket.
EXAMPLE
For the 2x2 we find
+-+-+ +-+-+ +-+-+ +-+-+ +-+-+
|X|X| |X|X| |X|X| |X| | |X| |
+-+-+ +-+-+ +-+-+ +-+-+ +-+-+
|X|X| |X| | | | | |X| | | |X|
+-+-+ +-+-+ +-+-+ +-+-+ +-+-+
+-+-+ +-+-+ +-+-+ +-+-+
|X|Y| |X| | |X| | |X|Y|
+-+-+ +-+-+ +-+-+ +-+-+
| | | |Y| | | |Y| |Z| |
+-+-+ +-+-+ +-+-+ +-+-+
so a(2) = 9.
CROSSREFS
KEYWORD
nonn
AUTHOR
Marko Riedel, Oct 04 2024
STATUS
approved