Necklaces-package {Necklaces}

Necklaces and Bracelets

Description

Tools to generate Necklaces, Bracelets, Lyndon words and de Bruijn sequences. The generation relies on integer partitions and uses the 'KStatistics' package. Methods used in the package refers to E. Di Nardo and G. Guarino (2022) <arXiv:2208.06855>.

Details

Using multi-index compositions, necklaces and bracelets are generated as well as Lyndon words and de Bruijn sequences. For multi-index compositions, this package refers to the kStatistics package.

Author(s)

Elvira Di Nardo [e-mail],
Giuseppe Guarino [e-mail]

Elvira Di Nardo [aut, cph], Giuseppe Guarino [aut, cre, cph]

Maintainer: Giuseppe Guarino <[e-mail]>

References

Di Nardo, E. (2014) On a symbolic representation of non-central Wishart random matrices with applications. Jour. Mult. Anal. Vol.125, 121–135. (https://arxiv.org/abs/1312.4395)

Di Nardo, E., and Guarino., G. (2022) kStatistics: Unbiased Estimates of Joint Cumulant Products from the Multivariate Faa Di Bruno's Formula. The R journal - In press. (https://arxiv.org/abs/2206.15348)

Di Nardo, E., and Guarino., G. (2022) Necklaces and bracelets in R - (https://arxiv.org/abs/2208.06855)

Flajolet, P., and Sedgewick, R. (2009) Analytic combinatorics. Cambridge University press.

Examples

# Sort the following list [2,2,3],[3,2,3],[1,2,3] 
#
lSort(list(c(2,2,3),c(3,2,3),c(1,2,3)))

# Generate the elements of the necklace in equivalence relation with 
# the input vector c(1,0,2,1) 
cNecklaces(c(1,0,2,1)) 

# The previous result in a compact form
cNecklaces(c(1,0,2,1),TRUE)

# Generate the elements of the bracelet in equivalence relation with 
# the input vector (1,0,2,1) 
cBracelets(c(1,0,2,1)) 

# The previous result in a compact form
cBracelets(c(1,0,2,1),TRUE)  

# Generate all the necklaces of the configuration (2,1,1) 
# corresponding to the vector (1,1,2,3)
fNecklaces(c(2,1,1))  

# The previous result in a compact form
fNecklaces(c(2,1,1),TRUE)

# The first value of the alphabet is set equal to zero 
fNecklaces(c(2,1,1),TRUE,0)

# Generate all the bracelets of the configuration (2,1,1) 
# corresponding to the vector (1,1,2,3)
fBracelets(c(2,1,1))  

# The previous result in a compact form
fBracelets(c(2,1,1),TRUE)

# The first value of the alphabet is set equal to zero
fBracelets(c(2,1,1),TRUE,0)

# Generate the list of all the representatives of all the necklaces 
# of length 4 over the alphabet {1,2}. 
Necklaces(4,2)  

# Generate the list of  all the representatives of all the necklaces 
# of length 5 over the alphabet {1,2,3}. 
Necklaces(5,3)  

# Generate the list of  all the representatives of all the necklaces 
# of length 5 over the alphabet {0,1,2}. 
Necklaces(5,3,0) 

# Generate the list of all the representatives of all the bracelets 
# of length 4 over the alphabet {1,2}. 
Bracelets(4,2)  

# Generate the list of  all the representatives of all the bracelets 
# of length 5 over the alphabet {1,2,3}. 
Bracelets(5,3)  

# Generate the list of  all the representatives of all the bracelets 
# of length 5 over the alphabet {0,1,2}. 
Bracelets(5,3,0)  

# Generate all the Lyndon words of length 5 over the alphabet 
# {1,2} 
LyndonW(5) 
# or equivalently 
LyndonW(5,2) 

# The previous result in a compact form 
LyndonW(5,2,TRUE)

# Generate all the Lyndon words of length 5 over the alphabet 
# {0,1}
LyndonW(5,2,TRUE,0)

# Generate the de Bruijn sequence of length 4 on the binary alphabet
# {0,1}
sBruijn(4) 
# or equivalently 
sBruijn(4,2) 

# Generate the de Bruijn sequence of length 2 over the alphabet {0,1,2} 
sBruijn(2,3)

# Generate the de Bruijn sequence of length 2 over the alphabet {1,2,3}
sBruijn(2,3,1)

# Generate the de Bruijn sequence of length 2 over the alphabet {1,2,3} 
# with a block separator. 
sBruijn(2,3,1,TRUE)