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| [[File:Partition necklaces by integer partition.svg|thumb|400px|Possible patterns of bracelets of length ''n''<br>corresponding to the ''k''-th [[Partition (number theory)|integer partition]]<br>([[Partition of a set|set partitions]] [[up to]] rotation and reflection)]]
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| [[File:Bracelets33.svg|thumb|300px|The 3 '''bracelets''' with 3 red and 3 green beads. The one in the middle is [[Chirality (mathematics)|chiral]], so there are 4 '''necklaces'''.<br><small>Compare box(6,9) in the triangle.</small>]]
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| [[File:Bracelets222.svg|thumb|300px|The 11 '''bracelets''' with 2 red, 2 yellow and 2 green beads. The leftmost one and the four rightmost ones are chiral, so there are 16 '''necklaces'''.<br><small>Compare box(6,7) in the triangle.</small>]]
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| [[File:Tantrix tiles ryg.svg|thumb|300px|16 [[Tantrix]] tiles, corresponding to the 16 '''necklaces''' with 2 red, 2 yellow and 2 green beads.]]
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| In [[combinatorics]], a ''k''-ary '''necklace''' of length ''n'' is an [[equivalence class]] of ''n''-character [[string (computer science)#Formal theory|string]]s over an [[alphabet (computer science)|alphabet]] of size ''k'', taking all [[circular shift|rotations]] as equivalent. It represents a structure with ''n'' circularly connected beads of up to ''k'' different colors.
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| A ''k''-ary '''bracelet''', also referred to as a '''turnover''' (or '''free''') '''necklace''', is a necklace such that strings may also be equivalent under reflection. That is, given two strings, if each is the reverse of the other then they belong to the same equivalence class. For this reason, a necklace might also be called a '''fixed necklace''' to distinguish it from a turnover necklace.
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| Technically, one may classify a necklace as an [[orbit (group theory)|orbit]] of the [[group action|action]] of the [[cyclic group]] on ''n''-character strings, and a bracelet as an orbit of the [[dihedral group]]'s action.
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| == Equivalence classes ==
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| === Number of necklaces ===
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| There are
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| :<math>N_k(n)={1\over n}\sum_{d\mid n}\varphi(d)k^{n/d}</math>
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| different ''k''-ary necklaces of length ''n'', where φ is the [[Euler's totient function]].<ref>http://mathworld.wolfram.com/Necklace.html</ref>
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| === Number of bracelets ===
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| There are
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| :<math>
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| B_k(n) =
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| \begin{cases}
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| {1\over 2}N_k(n) + {1\over 4}(k+1)k^{n/2} & \text{if }n\text{ is even} \\ \\
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| {1\over 2}N_k(n) + {1 \over 2}k^{(n+1)/2} & \text{if }n\text{ is odd}
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| \end{cases}
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| </math>
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| different ''k''-ary bracelets of length ''n'', where ''N''<sub>''k''</sub>(''n'') is the number of ''k''-ary necklaces of length ''n''.
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| == Examples ==
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| === Necklace example ===
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| If there are ''n'' beads, all unique, on a necklace joined at the ends, then the number of unique orderings on the necklace, after allowing for rotations, is ''n''!/''n'', for ''n'' > 0. This may also be expressed as (''n'' − 1)<nowiki>!</nowiki>. This number is less than the general case, which lacks the requirement that each bead must be unique.
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| An intuitive justification for this can be given. If there is a line of ''n'' unique objects ("beads"), the number of combinations would be ''n''!. If the ends are joined together, the number of combinations are divided by ''n'', as it is possible to rotate the string of ''n'' beads into ''n'' positions.
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| === Bracelet example ===
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| If there are ''n'' beads, all unique, on a bracelet joined at the ends, then the number of unique orderings on the bracelet, after allowing for rotations and reflection, is ''n''!/(2''n''), for ''n'' > 2. Note that this number is less than the general case of ''B<sub>n</sub>''(''n''), which lacks the requirement that each bead must be unique.
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| To explain this, one may begin with the count for a necklace. This number can be further divided by 2, because it is also possible to flip the bracelet over.
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| == Aperiodic necklaces ==
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| An '''aperiodic necklace''' of length ''n'' is an [[equivalence class]] of size ''n'', i.e., no two distinct rotations of a necklace from such class are equal.
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| According to [[Moreau's necklace-counting function]], there are
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| :<math>M_k(n)={1\over n}\sum_{d\mid n}\mu(d)k^{n/d}</math>
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| different ''k''-ary aperiodic necklaces of length ''n'', where ''μ'' is the [[Möbius function]].
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| Each aperiodic necklace contains a single [[Lyndon word]] so that Lyndon words form [[Equivalence Class Representative|representatives]] of aperiodic necklaces.
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| == Products of Necklaces ==
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| The limit of products of the numbers of the fixed necklaces of length n composed of beads of types <math>N_k(n)</math> :<math>N_k(n)={1\over n}\sum_{d\mid n}\varphi(d)k^{n/d}</math>:
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| :<math>\lim_{n \to \infty} \prod_{n=1}^n N_k(n)= \frac {k^n} {n!} 1(1+X)(1+X+X^2)\cdots(1+X+X^2+\cdots+X^{n-1}),</math>, <math>n \to \infty</math>,
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| where the coefficient of <math>X^k</math> in the expansion of the product
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| :<math>\prod_{m=1}^n\sum_{i=0}^{m-1}X^i=1(1+X)(1+X+X^2)\cdots(1+X+X^2+\cdots+X^{n-1})</math>
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| presents the number of [[permutations]] of n with k [[Inversion (discrete mathematics)|inversions]], expressed by a Mahonian number: {{OEIS link|A008302}} (See Gaichenkov link)
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| == See also ==
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| * [[Lyndon word]]
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| * [[Inversion (discrete mathematics)]]
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| * [[Necklace problem]]
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| * [[Necklace splitting problem]]
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| * [[Permutation]]
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| * [[Proofs of Fermat's little theorem#Proof by counting necklaces]]
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| * [[Forte number]], a representation of binary bracelets of length 12 used in [[atonal music]].
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| == References ==
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| {{reflist}}
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| == External links ==
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| * {{MathWorld|Necklace|Necklace}}
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| * [http://www.theory.csc.uvic.ca/~cos/inf/neck/NecklaceInfo.html Info on necklaces, Lyndon words, De Bruijn sequences]
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| [[Category:Combinatorics on words]]
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| [[Category:Enumerative combinatorics]]
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