A294686 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly four colors under translational symmetry, 1 <= k <= n.

0, 0, 6, 0, 260, 20720, 6, 5112, 1223136, 257706024, 48, 81876, 67769552, 54278580036, 44900438149488, 260, 1223396, 3731753700, 11681058472672, 38403264917970196, 131160169581733489616, 1200, 17815020, 207438938000, 2570217454576416, 33725471278376393424, 460532748521625850986660, 6467585568566200114362823920, 5106, 257706012, 11681057249536, 576229125971686224

Offset

1,3

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..820 (first 40 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=4 and S(n,k) Stirling numbers of the second kind.

T(n,k) = A184277(n,k) - 4*A184284(n,k) + 6*A184271(n,k) - 4. - Andrew Howroyd, Oct 05 2024

Example

Triangle begins:
    0;
    0,       6;
    0,     260,      20720;
    6,    5112,    1223136,      257706024;
   48,   81876,   67769552,    54278580036,    44900438149488;
  260, 1223396, 3731753700, 11681058472672, 38403264917970196, 131160169581733489616;
  ...

Prog

(PARI) T(n, m)=my(k=4); 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 A376824.

Cf. A294684, A294685, A294687, A294791, A294792, A294793, A294794. T(n,1) is A056284.

Cf. A184271, A184284, A184277.

Sequence in context: A362794 A336303 A111372 * A179936 A219952 A156444

Adjacent sequences: A294683 A294684 A294685 * A294687 A294688 A294689

Keyword

nonn,tabl,nice

Author

Marko Riedel, Nov 06 2017

Status

approved