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

1,5

Comments

Colors are not being permuted, i.e., Power Group Enumeration does not apply here.

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.

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.

Example

Triangle begins:
   0;
   0,      0;
   0,    300,      92680;
   0,  15750,   13794150,    8221452750;
  24, 510312, 1686135376, 4495236798162, 11696087875731720;
  ...

Prog

(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

Crossrefs

Main diagonal is A376825.

Cf. A294684, A294685, A294686, A294791, A294792, A294793, A294794. T(n,1) is A056285.

Sequence in context: A223350 A190880 A063935 * A015275 A051306 A151609

Adjacent sequences: A294684 A294685 A294686 * A294688 A294689 A294690

Keyword

nonn,tabl,nice

Author

Marko Riedel, Nov 06 2017

Status

approved