A294792 - OEIS

A294792

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

0, 0, 3, 1, 18, 345, 2, 136, 7254, 447156, 5, 946, 158355, 29032254, 5647919665, 18, 7324, 3580802, 1961010826, 1143822046786, 694881637942816, 43, 56450, 82968843, 136166703562, 238244961999013, 434202285631866206, 813943290958393433377, 126, 447138, 1960981598, 9651082393912, 50656925726930746, 276966813318877426118, 1557582240509759704455566

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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..35.

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=3. 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, A294793, A294794, A295197. T(n,1) is A056296.

Sequence in context: A283150 A307064 A120984 * A290317 A016480 A086156

Adjacent sequences: A294789 A294790 A294791 * A294793 A294794 A294795

KEYWORD

nonn,tabl

AUTHOR

Marko Riedel, Nov 08 2017

STATUS

approved