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Information on necklaces, unlabelled necklaces, Lyndon words, De Bruijn sequences

A k-ary necklace is an equivalence class of k-ary 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 k-ary alphabet:

Lk(n)   =  
1
n
 
__
\
/   
d | n
µ(n/d) k d ,           Nk(n)   =  
1
n
 
__
\
/   
d | n
Ø(n/d) k d ,
 
Lyndon words       Necklaces

The function µ(m) is the Möbius function: 0 if m is the product of non-distinct 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 k-ary De Bruijn sequence of order n is a circular k-ary string containing every k-ary string of length n as a substring exactly once. Thus a De Bruijn sequence has length kn. 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=a1..am, such that each ai is a necklace and ai > aj for all i < j. Similarily, every word can be factored uniquely into Lyndon words except that the factors satisfy ai > aj 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(n0,n1,...,nk-1) denote the number of necklaces composed of ni occurrences of the symbol i, for i = 0,...,k-1. Let the density of the necklace d = n1 + n2 + ... + nk-1 and n0 = n-d. It is known that

N(n0,n1,...,nk-1)   =  
1
n
  
   __
   \
   /   
j | gcd(n0,n1,...,nk-1)
Ø(j)
(
(n / j)!
(n0 / j)! (n1 / j)! ... (nk-1 / j)!
)

To get the number of fixed density necklaces with length n and density d, we sum over all possible values of n1,n2,...,nk-1

Nk(n,d)   =       
   __
   \
   /   
n1 + ... + nk-1 = d
N( n0, n1,...,nk-1)

The number of fixed density Lyndon words are counted similarily

L(n0,n1,...,nk-1)   =  
1
n
  
   __
   \
   /   
j | gcd(n0,n1,...,nk-1)
µ(j)
(
(n / j)!
(n0 / j)! (n1 / j)! ... (nk-1 / j)!
)

Lk(n,d)   =       
   __
   \
   /   
n1 + ... + nk-1 = d
L( n0, n1,...,nk-1)

For the binary case, these formulas simplify as follow:

L2(n,d)   =  
1
n
  
   __
   \
   /   
j | gcd(n,d)
µ(j)
(
n/j
d/j
)
          N2(n,d)   =  
1
n
  
   __
   \
   /   
j | gcd(n,d)
Ø(j)
(
n/j
d/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 0t. For example, when considering all necklaces with n=4 and k=2 with the restriction that there are no 00 (ie. 02) substrings we get the set: { 0101, 0111 ,1111 }. Notice that this is precisely the set of necklaces that start with 0j 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:

Bk(n)   =  

1
2
  Nk(n) +
1
4
  (k+1)kn/2         if n is even
1
2
  Nk(n) +
1
2
  k(n+1)/2         if n is odd
where Nk(n)   =  
1
n
 
__
\
/   
d | 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 n-1 white beads.

Information about the number of Lyndon words with given trace (sum of characters) mod q may be found here here.


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