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A295197
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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 any number of swappable colors.
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5
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1, 2, 9, 3, 43, 2387, 7, 587, 351773, 655089857, 12, 11703, 92197523, 2586209749712, 185543613289205809, 43, 352902, 37893376167, 18581620064907130, 28224967150633208580385, 106103186941524316132396201360, 127, 13639372, 22612848403571, 220019264470242220839, 8045720086273150473238405274, 851013076163633746725692124186472539, 218900758256599151027392153440612298654753249
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OFFSET
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1,2
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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.) Maximum number of colors is n * k.
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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) = Sum_{Q=1..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). 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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EXAMPLE
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The two-by-two with swappable colors has one monochrome coloring, four colorings with two colors, three colorings with three colors (determined by the color that appears twice) and one coloring with four colors.
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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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