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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.
11

%I #33 Jun 24 2018 16:00:01

%S 0,1,4,1,7,31,3,23,179,2107,3,55,1095,26271,671103,7,189,7327,350063,

%T 17896831,954459519,9,595,49939,4794087,490853415,52357746895,

%U 5744387279871,19,2101,349715,67115111,13743921631,2932032057731,643371380132743,144115188277194943,29,7315,2485591,954444607,390937468407,166799988703927,73201365371896619

%N 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.

%C 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.)

%D F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

%H Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2506511/">Burnside lemma and translational symmetries of the torus.</a>

%H Marko Riedel, <a href="/A294791/a294791_2.maple.txt">Maple code for sequences A294791, A294792, A294793, A294794.</a>

%F 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.

%e For the 2 X 2 grid and two colors we find T(2,2) = 4:

%e +---+ +---+ +---+ +---+

%e |X| | |X| | |X|X| |X| |

%e +-+-+ +-+-+ +-+-+ +-+-+

%e | | | | |X| | | | |X| |

%e +-+-+ +-+-+ +-+-+ +-+-+

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

%K nonn,tabl,nice

%O 1,3

%A _Marko Riedel_, Nov 08 2017