A294685 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry, 1 <= k <= n.

0, 0, 9, 2, 91, 2022, 9, 738, 43315, 2679246, 30, 5613, 950062, 174184755, 33887517990, 91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814, 258, 338259, 497812638, 816999710223, 1429469771994078, 2605213713043722909, 4883659745750360600262, 729, 2679228, 11765822365, 57906482267826, 303941554100145501

Offset: 1

Author

Marko Riedel, Nov 06 2017

Keywords

• nonn
• tabl
• nice

Comments

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

Examples

			Triangle begins:
   0;
   0,     9;
   2,    91,     2022;
   9,   738,    43315,     2679246;
  30,  5613,   950062,   174184755,   33887517990;
  91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814;
  ...
		

References

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..820 (first 40 rows)
• Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

Crossrefs

Main diagonal is A376823.

Cf. A184271, A184284, A294684, A294686, A294687, A294791, A294792, A294793, A294794. T(n,1) is A056283.

Programs

PARI

T(n,m)=6*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), 3, 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=3 and S(n,k) Stirling numbers of the second kind.

T(n,k) = A184284(n,k) - 3*A184271(n,k) + 3. - Andrew Howroyd, Oct 05 2024