OFFSET
6,3
COMMENTS
A necklace is a circular word considered up to rotation. A bracelet is a necklace considered up to rotation and reversal. For an integer partition lambda = (lambda_1, ..., lambda_k) of n, use k colors and require color i to occur lambda_i times. Thus lambda records the multiplicities of the colors, not the colors themselves; relabeling the colors does not give a new contribution for the same partition lambda.
This sequence counts only those fixed-content bracelets represented by a circular word X such that (1) the mirror image of X is not a rotation of X, and (2) some non-identity rotation fixes X. Equivalently, these are the chiral fixed-content bracelets whose rotational period is a proper divisor of n.
This sequence refines the partition-indexed bracelet count A213943 by splitting according to mirror-image behavior and rotational symmetry. The related cases give disjoint subcounts of A213943(n).
This class is sparse, because nontrivial rotational symmetry can occur only in composite lengths.
Can be computed by enumerating fixed-content necklaces with Sawada's algorithm and then classifying each necklace by mirror-image behavior and rotational period.
LINKS
Robert P. P. McKone, Table of n, a(n) for n = 6..139
Andrea Cutri, Python program for A396631.
Robert P. P. McKone, The fixed-content bracelets with n = 6 to n = 19 beads.
Robert P. P. McKone, The fixed-content bracelets with visual diagrams a(6) = 1.
Robert P. P. McKone, The fixed-content bracelets with visual diagrams a(8) = 4.
Robert P. P. McKone, The fixed-content bracelets with visual diagrams a(9) = 1.
Robert P. P. McKone, The fixed-content bracelets with visual diagrams a(10) = 22.
Robert P. P. McKone, The fixed-content bracelets with visual diagrams a(12) = 130.
Robert P. P. McKone, The fixed-content bracelets with visual diagrams a(14) = 809.
Robert P. P. McKone, The fixed-content bracelets with visual diagrams a(15) = 23.
Robert P. P. McKone, The fixed-content bracelets with visual diagrams a(16) = 5933.
FORMULA
a(n) = Sum_{lambda partition of n} b(lambda), where b(lambda) is the number of bracelets with color multiplicities lambda whose mirror image is not a rotation of the original necklace and whose rotational period is a proper divisor of n.
EXAMPLE
For n=6, the partition [2,2,2] means three colors, each used twice. The circular word 012012 is counted. Its distinct rotations are 012012, 120120, and 201201; rotation by 3 places fixes the word, so it has a nontrivial rotational symmetry and rotational period 3, a proper divisor of 6.
The mirror image of 012012 is 210210. The rotations of 210210 are 210210, 102102, and 021021, none of which is one of 012012, 120120, 201201. Thus the bracelet is different from its mirror image. Circular words obtained only by permuting the color names are not additional objects for the same partition [2,2,2]. This gives the single object counted by a(6) = 1.
MATHEMATICA
f[0] = 1;
f[n_Integer?Positive] := f[n] = Factorial[n] SeriesCoefficient[Product[1/(1 - x^j/Factorial[j]), {j, 1, n}], {x, 0, n}];
nl[n_Integer?Positive] := nl[n] = (1/n) Total[Table[EulerPhi[n/d] f[d], {d, Divisors[n]}]];
r[n_Integer?Positive] := r[n] = If[OddQ[n], With[{m = (n - 1)/2}, Total[Table[Binomial[m, j] f[j], {j, 0, m}]]], With[{h = n/2}, f[h] + (1/2) Total[Table[Binomial[h - 1, j] f[h - 1 - j]*(2^j + Boole[EvenQ[j]] Binomial[j, j/2]), {j, 0, h - 1}]]]];
a[n_Integer?Positive] := a[n] = (1/2) Total[Table[Total[Table[MoebiusMu[e/d] (nl[d] - r[d]), {d, Divisors[e]}]], {e, Select[Divisors[n], # < n &]}]];
Table[a[n], {n, 6, 41}] (* Robert P. P. McKone, Jun 16 2026 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrea Cutri, Jun 01 2026
EXTENSIONS
a(13) onward from Robert P. P. McKone, Jun 16 2026
STATUS
approved