A327693 Triangle read by rows: T(n,k) is the number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using exactly k different colored beads.

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 3, 5, 2, 0, 0, 4, 13, 9, 2, 0, 0, 9, 43, 50, 20, 3, 0, 0, 14, 116, 206, 127, 31, 3, 0, 0, 28, 335, 862, 772, 293, 51, 4, 0, 0, 48, 920, 3384, 4226, 2263, 580, 72, 4, 0, 0, 93, 2591, 13250, 22430, 16317, 5817, 1080, 105, 5, 0

Offset: 1

Author

Andrew Howroyd, Sep 22 2019

Keywords

• nonn
• tabl

Comments

Permuting the colors does not change the structure.

The definition requires that a necklace must not be equivalent to itself by permutation of colors and rotation (except for identity rotation). For example the length 2 necklace AB is excluded because a rotation of 1 gives BA and permutation of colors brings back to AB.

Examples

			Triangle begins:
  1;
  0,  0;
  0,  1,   0;
  0,  1,   1,    0;
  0,  3,   5,    2,    0;
  0,  4,  13,    9,    2,    0;
  0,  9,  43,   50,   20,    3,   0;
  0, 14, 116,  206,  127,   31,   3,  0;
  0, 28, 335,  862,  772,  293,  51,  4, 0;
  0, 48, 920, 3384, 4226, 2263, 580, 72, 4, 0;
  ...
T(6, 4) = 9: {aaabcd, aabacd, aabcad, aabbcd, aabcbd, aabcdb, aacbdb, ababcd, abacbd}. Compared with A107424 the patterns {abacad, aacbbd, abcabd, acabdb} are excluded.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Columns k=2..4 are A051841, A328740, A328741.

Row sums are A327696.

Partial row sums include A328742, A328743.

Cf. A324802 (not self-equivalent under reversal and rotations).

Cf. A107424, A152175.

Programs

PARI

R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, moebius(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
{ my(A=R(12)); for(n=1, #A, print(A[n, 1..n])) }