A294792 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry and swappable colors.

0, 0, 3, 1, 18, 345, 2, 136, 7254, 447156, 5, 946, 158355, 29032254, 5647919665, 18, 7324, 3580802, 1961010826, 1143822046786, 694881637942816, 43, 56450, 82968843, 136166703562, 238244961999013, 434202285631866206, 813943290958393433377, 126, 447138, 1960981598, 9651082393912, 50656925726930746, 276966813318877426118, 1557582240509759704455566

Offset: 1

Author

Marko Riedel, Nov 08 2017

Keywords

• nonn
• tabl

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.)

References

• F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Links

• Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

Crossrefs

Cf. A294684, A294685, A294686, A294687, A294791, A294793, A294794, A295197. T(n,1) is A056296.

Formula

T(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) with Q=3. 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.