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A327693
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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.
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7
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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
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OFFSET
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1,12
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(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])) }
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CROSSREFS
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Cf. A324802 (not self-equivalent under reversal and rotations).
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KEYWORD
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AUTHOR
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STATUS
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approved
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