A376749 Number of non-isomorphic colorings of a toroidal n X n grid using exactly four swappable colors.

0, 1, 874, 10741819, 1870851589562, 5465007068038102643, 269482732023591671431784330, 221537990355601030571170905795094315, 3007205014171762201565124875608675533096268906, 669557518440386985607930852942771727146772232484581602227, 2433673642945425535196140161775877796522974318753784273286700783313050

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 A294793.

Cf. A376747, A376748, A376808, A376824.

Formula

a(n) = (1/(n^2*4!))*(Sum_{sigma in S_4} 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..4} (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.