Information on necklaces, unlabelled necklaces, Lyndon words, De Bruijn sequences
A kary necklace is an equivalence class of kary
strings under rotation.
We take the lexicographically smallest such string as the representative
of each equivalence class and use this in the output of the program.
A Lyndon word is an aperiodic necklace representative.
The illustration on the right shows the 6 binary necklaces with 4 beads and
the corresponding equivalence classes of strings.
The representatives are colored and green indicates a Lyndon word.
The number of binary necklaces for n=1,2,...,15 is
2, 3, 4, 6, 8, 14, 20, 36, 60, 108, 188, 352, 632, 1182, 2192.
This is sequence
A000031(M0564) in
Neil J. Sloane's
database
of integer sequences.
The number of binary Lyndon words for n=1,2,...,15 is
2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182.
This is sequence
A001037(M0116) in
Neil J. Sloane's
database
of integer sequences.
There are explicit formulae for both of these sequences, given below
over a kary alphabet:
L_{k}(n) =

1 

n 



µ(n/d) k ^{d} ,


N_{k}(n) =

1 

n 



Ø(n/d) k
^{d} ,



 Lyndon words

  Necklaces

The function µ(m) is the Möbius function:
0 if m is the product of nondistinct primes, +1 if it is the
product of an even number of distinct primes, and 1 otherwise.
For example, µ(1) = 1,
µ(4) = 0,
µ(6) = +1, and
µ(7) = 1.
The function Ø(m)
is Euler's totient function, the number of
numbers
k in the range 1 < k < m
that are relatively prime to m.
For example Ø(1) = {1} = 1,
and Ø(6) =
{1,5} = 2.
A kary De Bruijn sequence of order n is a
circular kary string containing every kary
string of length n as a substring exactly once.
Thus a De Bruijn sequence has length k^{n}.
For n = 3 and k = 2, the lexicographically
smallest De Bruijn sequence is 00010111.
Every De Bruijn sequence corresponds to a Eulerian cycle on
a De Bruijn graph, one of which is shown below (for
n = 4 and k = 2).
At first glance De Bruijn sequences appear to have little to do with
necklaces or Lyndon words. However, Fredericksen has proven that the
lexicographic sequence of Lyndon words of lengths divisible by
n gives the lexicographically least De Bruijn sequence,
and that is the one output by our program.
The algorithm used is from the Combinatorial Generation book.
It is a recursive algorithm and generates necklaces in lexicographic
order, as does the first efficient necklace generation algorithm, the
iterative algorithm of Fredricksen, Maiorana, and Kelsen.
Lyndon and Necklace Factorizations
Given a string it is often useful to compute its necklace.
Such applications arise in several diverse areas such as graph
drawing, where it is used to help determine the symmetries of
a graph to be drawn in the plane.
To accomplish this task, we make use of necklace factorizations.
Every word a has a unique necklace factorization
a=a_{1}..a_{m},
such that each a_{i} is a necklace and
a_{i} > a_{j} for all
i < j.
Similarily, every word can be factored uniquely into Lyndon
words except that the factors satisfy
a_{i} > a_{j} for
all i < j.
The inequalities between words are with respect to lexicographic
order.
Duval has developed an elegent and efficient algorithm for factoring
a word that allows us to compute the necklace of a string.
Unlabelled Necklaces
Unlabelled necklaces are an equivalence classes of necklaces under
permutation of alphabet symbols.
Note that by permuting the alphabet symbols of a necklace and
then rotating the string into its lexicographically smallest position,
the result is a necklace representative.
Among all permutations of alphabet symbols
that resulting necklace that is lexicographically
smallest is used as the representative of an
unlabelled necklace.
For example, here is an equivalence class of unlabelled ternary
necklaces: { 011222, 022111, 001112, 002221, 000122, 000211 }.
The lexicographically smallest of these is 000122 and so it is
chosen as the representative.
The illustration on the right shows the 4 unlabelled binary necklaces with 4 beads
and the corresponding equivalence classes of necklaces.
The representatives are colored and green indicates an unlabelled Lyndon word.
The number of unlabelled necklaces for n=1,2,...,15 is
1, 2, 2, 4, 4, 8, 10, 20, 30, 56, 94, 180, 316, 596, 1096
This is sequence
A000013 in
Neil J. Sloane's
database
of integer sequences.
The number of unlabelled Lyndon words for n=1,2,...,15 is
1, 1, 1, 2, 3, 5, 9, 16, 28, 51, 93, 170, 315, 585, 1091
This is sequence
A000048 in
Neil J. Sloane's
database
of integer sequences.
Necklaces with Fixed Density
In many applications not all necklaces are required, but rather only
those of fixed density (the number of zeroes is fixed). In the more
general case, one may want a list of necklaces where the number of
occurences for every character is fixed.
To count fixed density necklaces we let
N(n_{0},n_{1},...,n_{k1}) denote the number of necklaces
composed of n_{i} occurrences of the symbol i, for
i = 0,...,k1. Let the density of the necklace
d = n_{1} + n_{2} + ... + n_{k1} and
n_{0} = nd. It is known that
N(n_{0},n_{1},...,n_{k1}) = 
1 

n 


__
 \
 /
 j  gcd(n_{0},n_{1},...,n_{k1})


Ø(j)

( 
(n / j)! 

(n_{0} / j)!
(n_{1} / j)! ...
(n_{k1} / j)! 

) 

To get the number of fixed density necklaces with length n and
density d, we sum over all possible values of
n_{1},n_{2},...,n_{k1}
N_{k}(n,d) = 


__
 \
 /
 n_{1} + ... + n_{k1} = d

 N( n_{0}, n_{1},...,n_{k1})

The number of fixed density Lyndon words are counted similarily
L(n_{0},n_{1},...,n_{k1}) = 
1 

n 


__
 \
 /
 j  gcd(n_{0},n_{1},...,n_{k1})


µ(j)

( 
(n / j)! 

(n_{0} / j)!
(n_{1} / j)! ...
(n_{k1} / j)! 

) 

L_{k}(n,d) = 


__
 \
 /
 n_{1} + ... + n_{k1} = d

 L( n_{0}, n_{1},...,n_{k1})

For the binary case, these formulas simplify as follow:
L_{2}(n,d) = 
1 

n 



µ(j)



N_{2}(n,d) = 
1 

n 



Ø(j)


Necklaces with forbidden sequences
Necklaces that do not contain a specified sequence as a substring are
known as necklaces with forbidden sequences. Currently we restrict the
forbidden sequence to be of the form 0^{t}. For example,
when
considering all necklaces with n=4 and k=2 with the
restriction that there are no 00 (ie. 0^{2})
substrings we get the set: { 0101, 0111 ,1111 }. Notice that this is
precisely the set of necklaces that start with 0^{j}
where j < t.
The number of necklaces with no 00 subsequence for n=1,2,...,15 is
1, 2, 2, 3, 3, 5, 5, 8, 10, 15, 19, 31, 41, 64, 94.
This is sequence
A000358(M0564) in
Neil J. Sloane's
database
of integer sequences.
Bracelets
A bracelet is a necklace that can be turned over.
The formula for a bracelet is:
where N_{k}(n) =

1 

n 



Ø(n/d) k
^{d} ,


For n = 1,2,...,20 the number of bracelets is
2, 3, 4, 6, 8, 13, 18, 30, 46, 78, 126, 224, 380, 687, 1224,
2250, 4112, 7685, 14310, 27012.
This is sequence
A000029(M0563) in
Neil J. Sloane's
database
of integer sequences.
For n = 1,2,...,20 the number of Lyndon bracelets is
2, 1, 2, 3, 6, 8, 16, 24, 42, 69, 124, 208, 378, 668, 1214,
2220, 4110, 7630, 14308, 26931.
This is sequence
A001371(M0115 in
Neil J. Sloane's
database
of integer sequences.
The number of nonisomorphic unit interval graphs is the same as the
number of bracelets with n black and n1
white beads.
Information about the number of Lyndon words with given trace
(sum of characters) mod q may be found here
here.
Programs available:
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