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A294793
Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly four colors under translational symmetry and swappable colors.
10
0, 0, 1, 0, 13, 874, 1, 235, 51075, 10741819, 2, 3437, 2823766, 2261625725, 1870851589562, 13, 51275, 155495153, 486711524815, 1600136051453135, 5465007068038102643, 50, 742651, 8643289534, 107092397450897, 1405227969932349726, 19188864521773558375127, 269482732023591671431784330, 221, 10741763, 486710971595, 24009547064476683
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.
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=4. 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
KEYWORD
nonn,tabl
AUTHOR
Marko Riedel, Nov 08 2017
STATUS
approved