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A294792
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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 three colors under translational symmetry and swappable colors.
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8
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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
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OFFSET
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1,3
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COMMENTS
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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.)
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REFERENCES
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F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
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LINKS
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FORMULA
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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