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.) Maximum number of colors is n * k.
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
LINKS
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequence A295197.
FORMULA
T(n,k) = Sum_{Q=1..n*k} (1/(n*k*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|k} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(k/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
The two-by-two with swappable colors has one monochrome coloring, four colorings with two colors, three colorings with three colors (determined by the color that appears twice) and one coloring with four colors.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Marko Riedel, Nov 16 2017
STATUS
approved