OFFSET
1,2
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 2*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 a non-identity rotation of X, (2) no rotation of X is a palindrome, and (3) no rotation other than the identity fixes X. Equivalently, these are the achiral fixed-content bracelets of full rotational period 2*n with no palindromic representative.
The condition that the mirror image is a non-identity rotation while no rotation is a palindrome forces the length to be even. Therefore this sequence lists the nonzero even-length values, indexed by half-length n, rather than inserting zeros for odd lengths.
This sequence refines the partition-indexed bracelet count A213943, restricted to even lengths, by splitting according to mirror-image behavior, palindromic representatives, and rotational symmetry. The related cases give disjoint subcounts of A213943(2*n).
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
Andrea Cutri, Python program for A396632.
FORMULA
a(n) = Sum_{lambda partition of 2*n} b(lambda), where b(lambda) is the number of bracelets with color multiplicities lambda whose mirror image is a non-identity rotation of the original necklace, with no palindromic rotation and rotational period 2*n.
EXAMPLE
For n=2 the underlying length is 4. The partition [3,1] means two colors, one used three times and one used once. It gives one counted bracelet, represented by the circular word 0001. Its mirror image 1000 is a non-identity rotation of 0001; none of the four rotations 0001, 0010, 0100, 1000 is a palindrome; and no non-identity rotation fixes the word. The word obtained by interchanging the two color names is not an additional object for the same partition [3,1].
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 0102. Its mirror image 2010 is a non-identity rotation of 0102; none of its rotations is a palindrome; and its rotational period is 4. Circular words obtained only by permuting the color names are not additional objects for the same partition [2,1,1]. Hence a(2) = 1 + 1 = 2.
CROSSREFS
KEYWORD
nonn,more,new
AUTHOR
Andrea Cutri, Jun 01 2026
EXTENSIONS
a(7) from Sean A. Irvine, Jul 06 2026
STATUS
approved