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A152176 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations and reflections. 15
1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 2, 1, 1, 7, 14, 11, 3, 1, 1, 8, 31, 33, 16, 3, 1, 1, 17, 82, 137, 85, 27, 4, 1, 1, 22, 202, 478, 434, 171, 37, 4, 1, 1, 43, 538, 1851, 2271, 1249, 338, 54, 5, 1, 1, 62, 1401, 6845, 11530, 8389, 3056, 590, 70, 5, 1, 1, 121, 3838, 26148 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Number of bracelet structures of length n using exactly k different colored beads. Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure. - Andrew Howroyd, Apr 06 2017

The number of achiral structures (A) is given in A140735 (odd n) and A293181 (even n).  The number of achiral structures plus twice the number of chiral pairs (A+2C) is given in A152175.  These can be used to determine A+C by taking half their average, as is done in the Mathematica program. - Robert A. Russell, Feb 24 2018

REFERENCES

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]

LINKS

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

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Tilman Piesk, Partition related number triangles

EXAMPLE

Triangle begins:

1;

1,  1;

1,  1,   1;

1,  3,   2,    1;

1,  3,   5,    2,    1;

1,  7,  14,   11,    3,    1;

1,  8,  31,   33,   16,    3,   1;

1, 17,  82,  137,   85,   27,   4,  1;

1, 22, 202,  478,  434,  171,  37,  4, 1;

1, 43, 538, 1851, 2271, 1249, 338, 54, 5, 1;

MATHEMATICA

Adn[d_, n_] := Adn[d, n] = Which[0==n, 1, 1==n, DivisorSum[d, x^# &],

  1==d, Sum[StirlingS2[n, k] x^k, {k, 0, n}],

  True, Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n - 1], x] x]];

Ach[n_, k_] := Ach[n, k] = Switch[k, 0, If[0==n, 1, 0], 1, If[n>0, 1, 0],

  (* else *) _, If[OddQ[n], Sum[Binomial[(n-1)/2, i] Ach[n-1-2i, k-1],

  {i, 0, (n-1)/2}], Sum[Binomial[n/2-1, i] (Ach[n-2-2i, k-1]

  + 2^i Ach[n-2-2i, k-2]), {i, 0, n/2-1}]]] (* achiral loops of length n, k colors *)

Table[(CoefficientList[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/(x n), x]

+ Table[Ach[n, k], {k, 1, n}])/2, {n, 1, 20}] // Flatten (* Robert A. Russell, Feb 24 2018 *)

PROG

(PARI) \\ see A056391 for Polya enumeration functions

T(n, k) = NonequivalentStructsExactly(DihedralPerms(n), k); \\ Andrew Howroyd, Oct 14 2017

CROSSREFS

Columns 2-6 are A056357, A056358, A056359, A056360, A056361.

Row sums are A084708.

Partial row sums include A000011, A056353, A056354, A056355, A056356.

Cf. A081720, A273891, A008277 (set partitions), A284949 (up to reflection), A152175 (up to rotation).

Sequence in context: A262311 A242950 A304972 * A152175 A134520 A188316

Adjacent sequences:  A152173 A152174 A152175 * A152177 A152178 A152179

KEYWORD

nonn,tabl

AUTHOR

Vladeta Jovovic, Nov 27 2008

STATUS

approved

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Last modified November 16 23:51 EST 2018. Contains 317275 sequences. (Running on oeis4.)