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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. 22
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

T(n,k)=pi_k(C_n) which is the number of non-equivalent partitions of the cycle on n vertices, with exactly k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. - Mohammad Hadi Shekarriz, Aug 21 2019

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

B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.

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

Tilman Piesk, Partition related number triangles

Mohammad Hadi Shekarriz, GAP Program

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

(PARI) \\ Ach is A304972 and R is A152175 as square matrices.

Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}

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)={(R(n) + Ach(n))/2}

{ my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 2019

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 A321620 A134520

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 December 13 06:26 EST 2019. Contains 329968 sequences. (Running on oeis4.)