

A295197


Triangle read by rows, 1 <= k <= n: T(n,k) = nonisomorphic colorings of a toroidal n X k grid using any number of swappable colors.


5



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

1,2


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.) Maximum number of colors is n * k.


REFERENCES

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


LINKS

Table of n, a(n) for n=1..28.
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequence A295197.


FORMULA

T(n,k) = Sum_{Q=1..n*k} (1/(n*k*Q!))*(Sum_{sigma in S_Q} Sum_{dn} Sum_{fk} phi(d) phi(f) [[forall j_l(sigma) > 0 : llcm(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.


EXAMPLE

The twobytwo 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.


CROSSREFS

Cf. A294791, A294792, A294793, A294794. T(n,1) is A084423.
Sequence in context: A088614 A162615 A275716 * A155163 A309929 A339799
Adjacent sequences: A295194 A295195 A295196 * A295198 A295199 A295200


KEYWORD

nonn,tabl


AUTHOR

Marko Riedel, Nov 16 2017


STATUS

approved



