OFFSET
1,2
COMMENTS
For an integer partition lambda = (lambda_1, ..., lambda_k) of 2*n+1, use k colors and require color i to occur lambda_i times. The profile lambda is fixed in weakly decreasing order; color relabelings are not additional objects for the same lambda. This is the strongly normal fixed-content convention.
Equivalently, a(n) counts aperiodic words w of length 2*n+1 satisfying w = reverse(w), summed over all strongly normal content profiles of that length.
The bracelet interpretation is obtained by taking the cyclic orbit of such a palindrome. Since the word is aperiodic, this orbit has full rotational period; since the word is a linear palindrome, its mirror image is a rotation of itself. Conversely, in odd length every full-period achiral fixed-content bracelet contains exactly one linear-palindrome representative.
Thus the sequence may also be read as the odd-length full-period achiral fixed-content bracelet class, but the primary definition is by linear palindromes rather than by the necklace-theoretic use of the word palindrome.
Even-length full-period achiral bracelets form different classes. For example, the length-4 word 1112 (profile [3,1]) has its reversal as a rotation but has no linear-palindrome representative; such no-linear-palindrome cases belong to the separate class S0,ns, published as A396632.
The sequence is one class in a disjoint refinement of strongly normal fixed-content bracelet counts by mirror-image behavior, linear-palindrome representatives, and rotational symmetry. This convention is related to, but not identical with, interchangeable-color bracelet totals.
Can be computed either from the palindrome formula below or by enumerating fixed-content necklaces with Sawada's algorithm and then classifying each necklace by mirror-image behavior, linear-palindrome representatives, and rotational period.
LINKS
Andrea Cutri, Table of n, a(n) for n = 1..300.
Andrea Cutri, Python program.
FORMULA
a(n) = Sum_{d | 2*n+1} mu((2*n+1)/d) * T(d), where T(m) = Sum_{lambda partition of m with exactly one odd part} ((m-1)/2)! / Product_i floor(lambda_i/2)! counts all linear palindromes of odd length m in the strongly normal fixed-content convention, and mu is the Moebius function.
a(n) = Sum_{lambda partition of 2*n+1} b(lambda), where b(lambda) is the number of bracelets with color multiplicities lambda whose mirror image is a rotation of the original necklace, with a linear-palindrome representative and rotational period 2*n+1.
EXAMPLE
For n=1 the underlying length is 3. The partition [2,1] means one color used twice and another color used once. It gives the single counted object, represented by the aperiodic linear palindrome 010. Equivalently, the cyclic orbit represented by 001 is achiral, has full rotational period, and contains 010 as its linear-palindrome representative.
The partitions [3] and [1,1,1] contribute nothing for n=1: the palindrome 000 is not aperiodic, and no linear palindrome has three distinct letters. Circular words obtained only by permuting the color names are not additional objects for the same partition. Hence a(1) = 1.
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Andrea Cutri, Jul 01 2026
STATUS
approved