A294687 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly five colors under translational symmetry, 1 <= k <= n.
0, 0, 0, 0, 300, 92680, 0, 15750, 13794150, 8221452750, 24, 510312, 1686135376, 4495236798162, 11696087875731720, 300, 13794450, 193054017440, 2425003938178050, 30852000867277668428, 403564024914127655401650, 2400, 343501500, 21664357535320, 1317601563731383350, 82985159653854019928352, 5411356249329837891442095560
Offset: 1
Examples
Triangle begins: 0; 0, 0; 0, 300, 92680; 0, 15750, 13794150, 8221452750; 24, 510312, 1686135376, 4495236798162, 11696087875731720; ...
References
- F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..325 (first 25 rows).
- Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Crossrefs
Programs
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PARI
T(n,m)=my(k=5); k!*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), k, 2) ))/(n*m) \\ Andrew Howroyd, Oct 05 2024
Formula
T(n,k) = (Q!/(n*k))*(Sum_{d|n} Sum_{f|k} phi(d) phi(f) S(gcd(d,f)*(n/d)*(k/f), Q)) with Q=5 and S(n,k) Stirling numbers of the second kind.
Comments