A294794 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly five colors under translational symmetry and swappable colors.

0, 0, 0, 0, 3, 775, 0, 145, 115100, 68522769, 1, 4281, 14051164, 37460388596, 97467398965031, 3, 115381, 1608801153, 20208371722051, 257100007425866689, 3363033541015148835823, 20, 2863227, 180536313547, 10980013072900632, 691542997115450167856, 45094635411084308447578413, 3020745549854628001139950947779, 136, 68522707

Offset

1,5

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

Table of n, a(n) for n=1..30.

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

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=5. 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.

Crossrefs

Cf. A294684, A294685, A294686, A294687, A294791, A294792, A294793, A295197. T(n,1) is A056298.

Sequence in context: A172895 A259369 A259371 * A293252 A341567 A287695

Adjacent sequences: A294791 A294792 A294793 * A294795 A294796 A294797

Keyword

nonn,tabl

Author

Marko Riedel, Nov 08 2017

Status

approved