A294791 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry and swappable colors.

0, 1, 4, 1, 7, 31, 3, 23, 179, 2107, 3, 55, 1095, 26271, 671103, 7, 189, 7327, 350063, 17896831, 954459519, 9, 595, 49939, 4794087, 490853415, 52357746895, 5744387279871, 19, 2101, 349715, 67115111, 13743921631, 2932032057731, 643371380132743, 144115188277194943, 29, 7315, 2485591, 954444607, 390937468407, 166799988703927, 73201365371896619

Offset: 1

Author

Marko Riedel, Nov 08 2017

Keywords

• nonn
• tabl
• nice

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.)

Examples

			For the 2 X 2 grid and two colors we find T(2,2) = 4:
  +---+  +---+  +---+  +---+
  |X| |  |X| |  |X|X|  |X| |
  +-+-+  +-+-+  +-+-+  +-+-+
  | | |  | |X|  | | |  |X| |
  +-+-+  +-+-+  +-+-+  +-+-+
		

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 sequences A294791, A294792, A294793, A294794.

Crossrefs

Cf. A152175, A294684, A294685, A294686, A294687, A294792, A294793, A294794, A295197. T(n,1) is A056295.

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=2. 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.