A376748 Number of non-isomorphic colorings of a toroidal n X n grid using exactly three swappable colors.

0, 3, 345, 447156, 5647919665, 694881637942816, 813943290958393433377, 8941884948534360647405572800, 912400181570021638669407666368774097, 858962534553352212055863239761275173880606456, 7425662396340624836407113113710889289196975262054947345, 587417576454184723055270940786413231085263155884260701824558793960

Offset: 1

Author

Marko Riedel, Oct 03 2024

Keywords

• nonn

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.

Crossrefs

Main diagonal of A294792.

Cf. A376747, A376749, A376808, A376823.

Formula

a(n) = (1/(n^2*3!))*(Sum_{sigma in S_3} 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..3} (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.