A396630 Number of fixed-content bracelets of length n, summed over all integer partitions of n, that are different from their mirror image and have no nontrivial rotational symmetry.

0, 0, 1, 4, 22, 125, 809, 5929, 48336, 443762, 4469217, 49647760, 597806704, 7809581207, 109524955614

Offset

1,4

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 for which (1) the mirror image of the circular word is not a rotation of the original word, and (2) no rotation other than the identity fixes the word. In the usual terminology, these are the chiral fixed-content bracelets with full rotational period 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).

For the computed terms n = 3..12, this is the largest of the corresponding mirror-image/rotational-symmetry subcases.

For n = 0, 1, 2 there are no bracelets satisfying the two defining conditions, so the sequence is displayed starting with the first nonzero term a(3) = 1.

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

Table of n, a(n) for n=1..15.

Andrea Cutri, Python program.

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 n.

Example

For n=4, the partition [1,1,1,1] means four colors, each used once. There are three counted bracelets with these multiplicities, represented for example by the circular words 0123, 0132, and 0213. The word 0123 is counted because its mirror image 3210 is not a rotation of 0123, and no non-identity rotation fixes it.
The partition [2,1,1] means one color is used twice and two other colors are used once. It gives one additional counted bracelet, represented by 0012. The words 1102 and 2201 are not additional counted objects here: they are obtained from 0012 by relabeling the colors, and the partition [2,1,1] records only the multiplicities. Hence a(4) = 3 + 1 = 4.

Crossrefs

Cf. A005651, A072605, A213943, A396631, A396632.

Sequence in context: A136777 A056625 A116428 * A121187 A011789 A047039

Adjacent sequences: A396627 A396628 A396629 * A396631 A396632 A396633

Keyword

nonn,more

Author

Andrea Cutri, Jun 01 2026

Extensions

a(13)-a(15) from Sean A. Irvine, Jun 22 2026

Status

approved