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
Marko Riedel, Bracelets with swappable colors classified by the distribution of colors, Power Group Enumeration algorithm
Marko Riedel, Maple code for the number of bracelets with some number of swappable colors by Power Group Enumeration
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
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
KEYWORD
nonn,tabl
AUTHOR
Vladeta Jovovic, Nov 27 2008
STATUS
approved