Lyndon words

class sage.combinat.lyndon_word.LyndonWord(data, check=True)

Bases: sage.combinat.words.word.FiniteWord_list

sage.combinat.lyndon_word.LyndonWords(e=None, k=None)

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.

INPUT:

• no input at all

or

• e - integer, size of alphabet
• k - integer, length of the words

or

• e - a composition

OUTPUT:

A combinatorial class of Lyndon words.

EXAMPLES:

sage: LyndonWords()
Lyndon words

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]

class sage.combinat.lyndon_word.LyndonWords_class(category=None, *keys, **opts)

Bases: sage.combinat.words.words.Words_all

class sage.combinat.lyndon_word.LyndonWords_evaluation(e)

Bases: sage.combinat.combinat.CombinatorialClass

cardinality()

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

class sage.combinat.lyndon_word.LyndonWords_nk(n, k)

Bases: sage.combinat.words.words.FiniteWords_length_k_over_OrderedAlphabet

cardinality()

TESTS:

sage: [ LyndonWords(3,i).cardinality() for i in range(1, 11) ]
[3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880]

sage.combinat.lyndon_word.StandardBracketedLyndonWords(n, k)

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]]

class sage.combinat.lyndon_word.StandardBracketedLyndonWords_nk(n, k)

Bases: sage.combinat.combinat.CombinatorialClass

cardinality()

EXAMPLES:

sage: StandardBracketedLyndonWords(3, 3).cardinality()
8
sage: StandardBracketedLyndonWords(3, 4).cardinality()
18

sage.combinat.lyndon_word.standard_bracketing(lw)

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]]