Returns the combinatorial class of Lyndon words.
A Lyndon word is a word that is lexicographically less than all of
its rotations. Equivalently, whenever
is split into two non-empty
substrings,
is lexicographically less than the right substring.
EXAMPLES: If e is an integer, then e specifies the length of the alphabet; k must also be specified in this case.
sage: LW = LyndonWords(3,3); LW
Lyndon words from an alphabet of size 3 of length 3
sage: LW.first()
word: 112
sage: LW.last()
word: 233
sage: LW.random_element()
word: 112
sage: LW.cardinality()
8
If e is a (weak) composition, then it returns the class of Lyndon words that have evaluation e.
sage: LyndonWords([2, 0, 1]).list()
[word: 113]
sage: LyndonWords([2, 0, 1, 0, 1]).list()
[word: 1135, word: 1153, word: 1315]
sage: LyndonWords([2, 1, 1]).list()
[word: 1123, word: 1132, word: 1213]
TESTS:
sage: LW33 = LyndonWords(3,3)
sage: all([lw in LyndonWords() for lw in LW33])
True
EXAMPLES:
sage: [1,2,1,2] in LyndonWords([2,2])
False
sage: [1,1,2,2] in LyndonWords([2,2])
True
sage: all([ lw in LyndonWords([2,1,3,1]) for lw in LyndonWords([2,1,3,1])])
True
TESTS:
sage: LW21 = LyndonWords([2,1]); LW21
Lyndon words with evaluation [2, 1]
sage: LW21 == loads(dumps(LW21))
True
An iterator for the Lyndon words with evaluation e.
EXAMPLES:
sage: LyndonWords([1]).list() #indirect doctest
[word: 1]
sage: LyndonWords([2]).list() #indirect doctest
[]
sage: LyndonWords([3]).list() #indirect doctest
[]
sage: LyndonWords([3,1]).list() #indirect doctest
[word: 1112]
sage: LyndonWords([2,2]).list() #indirect doctest
[word: 1122]
sage: LyndonWords([1,3]).list() #indirect doctest
[word: 1222]
sage: LyndonWords([3,3]).list() #indirect doctest
[word: 111222, word: 112122, word: 112212]
sage: LyndonWords([4,3]).list() #indirect doctest
[word: 1111222, word: 1112122, word: 1112212, word: 1121122, word: 1121212]
TESTS:
sage: repr(LyndonWords([2,1,1]))
'Lyndon words with evaluation [2, 1, 1]'
Returns the number of Lyndon words with the evaluation e.
EXAMPLES:
sage: LyndonWords([]).cardinality()
0
sage: LyndonWords([2,2]).cardinality()
1
sage: LyndonWords([2,3,2]).cardinality()
30
Check to make sure that the count matches up with the number of Lyndon words generated.
sage: comps = [[],[2,2],[3,2,7],[4,2]]+Compositions(4).list()
sage: lws = [ LyndonWords(comp) for comp in comps]
sage: all( [ lw.cardinality() == len(lw.list()) for lw in lws] )
True
TESTS:
sage: LW33 = LyndonWords(3,3)
sage: all([lw in LW33 for lw in LW33])
True
TESTS:
sage: LW23 = LyndonWords(2,3); LW23
Lyndon words from an alphabet of size 2 of length 3
sage: LW23== loads(dumps(LW23))
True
TESTS:
sage: LyndonWords(3,3).list() # indirect doctest
[word: 112, word: 113, word: 122, word: 123, word: 132, word: 133, word: 223, word: 233]
TESTS:
sage: repr(LyndonWords(2, 3))
'Lyndon words from an alphabet of size 2 of length 3'
TESTS:
sage: [ LyndonWords(3,i).cardinality() for i in range(1, 11) ]
[3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880]
Returns the combinatorial class of standard bracketed Lyndon words from [1, ..., n] of length k. These are in one to one correspondence with the Lyndon words and form a basis for the subspace of degree k of the free Lie algebra of rank n.
EXAMPLES:
sage: SBLW33 = StandardBracketedLyndonWords(3,3); SBLW33
Standard bracketed Lyndon words from an alphabet of size 3 of length 3
sage: SBLW33.first()
[1, [1, 2]]
sage: SBLW33.last()
[[2, 3], 3]
sage: SBLW33.cardinality()
8
sage: SBLW33.random_element()
[1, [1, 2]]
TESTS:
sage: SBLW = StandardBracketedLyndonWords(3, 2)
sage: SBLW == loads(dumps(SBLW))
True
EXAMPLES:
sage: StandardBracketedLyndonWords(3, 3).list()
[[1, [1, 2]],
[1, [1, 3]],
[[1, 2], 2],
[1, [2, 3]],
[[1, 3], 2],
[[1, 3], 3],
[2, [2, 3]],
[[2, 3], 3]]
TESTS:
sage: repr(StandardBracketedLyndonWords(3, 3))
'Standard bracketed Lyndon words from an alphabet of size 3 of length 3'
EXAMPLES:
sage: StandardBracketedLyndonWords(3, 3).cardinality()
8
sage: StandardBracketedLyndonWords(3, 4).cardinality()
18
Returns the standard bracketing of a Lyndon word lw.
EXAMPLES:
sage: import sage.combinat.lyndon_word as lyndon_word
sage: map( lyndon_word.standard_bracketing, LyndonWords(3,3) )
[[1, [1, 2]],
[1, [1, 3]],
[[1, 2], 2],
[1, [2, 3]],
[[1, 3], 2],
[[1, 3], 3],
[2, [2, 3]],
[[2, 3], 3]]