A376747 Number of non-isomorphic colorings of a toroidal n X n grid using exactly two swappable colors.

0, 4, 31, 2107, 671103, 954459519, 5744387279871, 144115188277194943, 14925010118699132241919, 6338253001141180784480847871, 10985355337065420437221545952731135, 77433143050453552574875182200691073835007, 2213872302702432822841084717014014514981767643135, 256208234097415541381052629523530965709132732687965552639

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 A294791.

Cf. A376748, A376749, A376808, A376822.

Formula

a(n) = (1/(n^2*2!))*(Sum_{sigma in S_2} 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..2} (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.