A056305 -id:A056305 - cp's OEIS frontend

A107424 Triangle read by rows: T(n, k) is the number of primitive (period n) n-bead necklace structures with k different colors. Only includes structures that contain all k colors.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 2, 1, 0, 5, 17, 13, 3, 1, 0, 9, 43, 50, 20, 3, 1, 0, 16, 124, 220, 136, 36, 4, 1, 0, 28, 338, 866, 773, 296, 52, 4, 1, 0, 51, 941, 3435, 4280, 2303, 596, 78, 5, 1, 0, 93, 2591, 13250, 22430, 16317, 5817, 1080, 105, 5, 1, 0, 170, 7234, 51061

Offset: 1

David Wasserman, May 26 2005

This classification is concerned with which beads are the same color, not with the colors themselves, so bbabcd is the same structure as aabacd. Cyclic permutations are also the same structure, e.g. abacda is also the same structure. However, order matters: the reverse of aabacd is equivalent to aabcad, which is also on the list.

			T(6, 4) = 13: {aaabcd, aabacd, aabcad, abacad, aabbcd, aabcbd, aabcdb, aacbbd, aacbdb, ababcd, abacbd, acabdb, abcabd}.
From _Andrew Howroyd_, Apr 09 2017 (Start)
Triangle starts:
1
0  1
0  1   1
0  2   2    1
0  3   5    2    1
0  5  17   13    3    1
0  9  43   50   20    3   1
0 16 124  220  136   36   4  1
0 28 338  866  773  296  52  4 1
0 51 941 3435 4280 2303 596 78 5 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns 2-6 are A056303, A056304, A056305, A056306, A056307.
Partial row sums include A000048, A002075, A056300, A056301, A056302.
Row sums are A276547.
Cf. A074650, A152175, A276543, A137651, A276544.

A[d_, n_] := A[d, n] = Which[n == 0, 1, n == 1, DivisorSum[d, x^# &], d == 1, Sum[StirlingS2[n, k] x^k, {k, 0, n}], True, Expand[A[d, 1] A[d, n-1] + D[A[d, n-1], x] x]];
B[n_, k_] := Coefficient[DivisorSum[n, EulerPhi[#] A[#, n/#]&]/n/x, x, k];
T[n_, k_] := DivisorSum[n, MoebiusMu[n/#] B[#, k]&];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jun 06 2018, after Andrew Howroyd and Robert A. Russell *)

\\ here R(n) is A152175 as square matrix.
R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n) = {my(M=R(n)); matrix(n, n, i, k, sumdiv(i, d, moebius(i/d)*M[d,k]))}
{ my(A=T(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 09 2020

T(n, k) = Sum_{d|n} mu(n/d) * A152175(d, k). - Andrew Howroyd, Apr 09 2017

A328741 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using exactly four different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 2, 9, 50, 206, 862, 3384, 13250, 50852, 194810, 741884, 2823712, 10735998, 40843370, 155483873, 592614050, 2261585918, 8643288808, 33080772468, 126797503250, 486710415710, 1870851589552, 7201012694120, 27752927349880, 107092389688830, 413729680838330

Offset: 1

Andrew Howroyd, Oct 26 2019

nonn

Permuting the colors does not change the structure.

			For n=6, the 9 necklace structures are: aaabcd, aabacd, aabcad, aabbcd, aabcbd, aabcdb, aacbdb, ababcd, abacbd.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column 4 of A327693.

Cf. A056297, A056305, A328740, A328743.

a(p) = A056297(p) = A056305(p) for prime p.

A056368 Number of primitive (period n) bracelet structures using exactly four different colored beads.

Original entry on oeis.org

0, 0, 0, 1, 2, 11, 33, 136, 478, 1849, 6845, 26136, 98406, 373977, 1416249, 5380770, 20440250, 77794939, 296384565, 1131009781, 4321964735, 16541268223, 63400061153, 243358777620, 935431121460, 3600520732809

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Column 4 of A276543.

Cf. A056305.

A056363(n)-A056362(n).

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A107424 Triangle read by rows: T(n, k) is the number of primitive (period n) n-bead necklace structures with k different colors. Only includes structures that contain all k colors.

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A328741 Number of n-bead necklace structures which are not self-equivalent under a nonzero rotation using exactly four different colored beads.

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A056368 Number of primitive (period n) bracelet structures using exactly four different colored beads.

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