Word classes#

AUTHORS:

  • Arnaud Bergeron

  • Amy Glen

  • Sébastien Labbé

  • Franco Saliola

class sage.combinat.words.word.FiniteWord_callable(parent, callable, length=None)[source]#

Bases: WordDatatype_callable, FiniteWord_class

Finite word represented by a callable.

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

EXAMPLES:

sage: f = lambda n : 3 if n > 8 else 6
sage: w = Word(f, length=30, caching=False)
sage: w
word: 666666666333333333333333333333
sage: w.is_symmetric()
True

>>> from sage.all import *
>>> f = lambda n : Integer(3) if n > Integer(8) else Integer(6)
>>> w = Word(f, length=Integer(30), caching=False)
>>> w
word: 666666666333333333333333333333
>>> w.is_symmetric()
True
class sage.combinat.words.word.FiniteWord_callable_with_caching(parent, callable, length=None)[source]#

Bases: WordDatatype_callable_with_caching, FiniteWord_class

Finite word represented by a callable (with caching).

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

EXAMPLES:

sage: f = lambda n : n % 3
sage: w = Word(f, length=32)
sage: w
word: 01201201201201201201201201201201
sage: w.border()
word: 01201201201201201201201201201

>>> from sage.all import *
>>> f = lambda n : n % Integer(3)
>>> w = Word(f, length=Integer(32))
>>> w
word: 01201201201201201201201201201201
>>> w.border()
word: 01201201201201201201201201201
class sage.combinat.words.word.FiniteWord_char[source]#

Bases: WordDatatype_char, FiniteWord_class

Finite word represented by an array unsigned char * (i.e. integers between 0 and 255).

For any word w, type w.<TAB> to see the functions that can be applied to w.

EXAMPLES:

sage: W = Words(range(20))

sage: w = W(list(range(1, 10)) * 2)
sage: type(w)
<class 'sage.combinat.words.word.FiniteWord_char'>
sage: w
word: 123456789123456789

sage: w.is_palindrome()
False
sage: (w*w[::-1]).is_palindrome()
True
sage: (w[:-1:]*w[::-1]).is_palindrome()
True

sage: w.is_lyndon()
False
sage: W(list(range(10)) + [10, 10]).is_lyndon()
True

sage: w.is_square_free()
False
sage: w[:-1].is_square_free()
True

sage: u = W([randint(0,10) for i in range(10)])
sage: (u*u).is_square()
True
sage: (u*u*u).is_cube()
True

sage: len(w.factor_set())
127
sage: w.rauzy_graph(5)                                                          # needs sage.graphs
Looped digraph on 9 vertices

sage: u = W([1,2,3])
sage: w.first_occurrence(u)
0
sage: w.first_occurrence(u, start=1)
9

>>> from sage.all import *
>>> W = Words(range(Integer(20)))

>>> w = W(list(range(Integer(1), Integer(10))) * Integer(2))
>>> type(w)
<class 'sage.combinat.words.word.FiniteWord_char'>
>>> w
word: 123456789123456789

>>> w.is_palindrome()
False
>>> (w*w[::-Integer(1)]).is_palindrome()
True
>>> (w[:-Integer(1):]*w[::-Integer(1)]).is_palindrome()
True

>>> w.is_lyndon()
False
>>> W(list(range(Integer(10))) + [Integer(10), Integer(10)]).is_lyndon()
True

>>> w.is_square_free()
False
>>> w[:-Integer(1)].is_square_free()
True

>>> u = W([randint(Integer(0),Integer(10)) for i in range(Integer(10))])
>>> (u*u).is_square()
True
>>> (u*u*u).is_cube()
True

>>> len(w.factor_set())
127
>>> w.rauzy_graph(Integer(5))                                                          # needs sage.graphs
Looped digraph on 9 vertices

>>> u = W([Integer(1),Integer(2),Integer(3)])
>>> w.first_occurrence(u)
0
>>> w.first_occurrence(u, start=Integer(1))
9
class sage.combinat.words.word.FiniteWord_iter(parent, iter, length=None)[source]#

Bases: WordDatatype_iter, FiniteWord_class

Finite word represented by an iterator.

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

EXAMPLES:

sage: w = Word(iter(range(10)), caching=False)
sage: w
word: 0123456789
sage: w.finite_differences()
word: 111111111

>>> from sage.all import *
>>> w = Word(iter(range(Integer(10))), caching=False)
>>> w
word: 0123456789
>>> w.finite_differences()
word: 111111111
class sage.combinat.words.word.FiniteWord_iter_with_caching(parent, iter, length=None)[source]#

Bases: WordDatatype_iter_with_caching, FiniteWord_class

Finite word represented by an iterator (with caching).

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

EXAMPLES:

sage: w = Word(iter('abcdef'))
sage: w.conjugate(2)
word: cdefab

>>> from sage.all import *
>>> w = Word(iter('abcdef'))
>>> w.conjugate(Integer(2))
word: cdefab
class sage.combinat.words.word.FiniteWord_list[source]#

Bases: WordDatatype_list, FiniteWord_class

Finite word represented by a Python list.

For any word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

EXAMPLES:

sage: w = Word(range(10))
sage: w.iterated_right_palindromic_closure()
word: 0102010301020104010201030102010501020103...

>>> from sage.all import *
>>> w = Word(range(Integer(10)))
>>> w.iterated_right_palindromic_closure()
word: 0102010301020104010201030102010501020103...
class sage.combinat.words.word.FiniteWord_morphic(parent, morphism, letter, coding=None, length=+Infinity)[source]#

Bases: WordDatatype_morphic, FiniteWord_class

Finite morphic word.

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

EXAMPLES:

sage: m = WordMorphism("a->ab,b->")
sage: w = m.fixed_point("a")
sage: w
word: ab

>>> from sage.all import *
>>> m = WordMorphism("a->ab,b->")
>>> w = m.fixed_point("a")
>>> w
word: ab
class sage.combinat.words.word.FiniteWord_str[source]#

Bases: WordDatatype_str, FiniteWord_class

Finite word represented by a Python str.

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

EXAMPLES:

sage: w = Word('abcdef')
sage: w.is_square()
False

>>> from sage.all import *
>>> w = Word('abcdef')
>>> w.is_square()
False
class sage.combinat.words.word.FiniteWord_tuple[source]#

Bases: WordDatatype_tuple, FiniteWord_class

Finite word represented by a Python tuple.

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

EXAMPLES:

sage: w = Word(())
sage: w.is_empty()
True

>>> from sage.all import *
>>> w = Word(())
>>> w.is_empty()
True
class sage.combinat.words.word.InfiniteWord_callable(parent, callable, length=None)[source]#

Bases: WordDatatype_callable, InfiniteWord_class

Infinite word represented by a callable.

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

Infinite words behave like a Python list : they can be sliced using square braquets to define for example a prefix or a factor.

EXAMPLES:

sage: w = Word(lambda n:n, caching=False)
sage: w
word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,...
sage: w.iterated_right_palindromic_closure()
word: 0102010301020104010201030102010501020103...

>>> from sage.all import *
>>> w = Word(lambda n:n, caching=False)
>>> w
word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,...
>>> w.iterated_right_palindromic_closure()
word: 0102010301020104010201030102010501020103...
class sage.combinat.words.word.InfiniteWord_callable_with_caching(parent, callable, length=None)[source]#

Bases: WordDatatype_callable_with_caching, InfiniteWord_class

Infinite word represented by a callable (with caching).

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

Infinite words behave like a Python list : they can be sliced using square braquets to define for example a prefix or a factor.

EXAMPLES:

sage: w = Word(lambda n:n)
sage: factor = w[4:13]
sage: factor
word: 4,5,6,7,8,9,10,11,12

>>> from sage.all import *
>>> w = Word(lambda n:n)
>>> factor = w[Integer(4):Integer(13)]
>>> factor
word: 4,5,6,7,8,9,10,11,12
class sage.combinat.words.word.InfiniteWord_iter(parent, iter, length=None)[source]#

Bases: WordDatatype_iter, InfiniteWord_class

Infinite word represented by an iterable.

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

Infinite words behave like a Python list : they can be sliced using square braquets to define for example a prefix or a factor.

EXAMPLES:

sage: from itertools import chain, cycle
sage: w = Word(chain('letsgo', cycle('forever')), caching=False)
sage: w
word: letsgoforeverforeverforeverforeverforeve...
sage: prefix = w[:100]
sage: prefix
word: letsgoforeverforeverforeverforeverforeve...
sage: prefix.is_lyndon()
False

>>> from sage.all import *
>>> from itertools import chain, cycle
>>> w = Word(chain('letsgo', cycle('forever')), caching=False)
>>> w
word: letsgoforeverforeverforeverforeverforeve...
>>> prefix = w[:Integer(100)]
>>> prefix
word: letsgoforeverforeverforeverforeverforeve...
>>> prefix.is_lyndon()
False
class sage.combinat.words.word.InfiniteWord_iter_with_caching(parent, iter, length=None)[source]#

Bases: WordDatatype_iter_with_caching, InfiniteWord_class

Infinite word represented by an iterable (with caching).

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

Infinite words behave like a Python list : they can be sliced using square braquets to define for example a prefix or a factor.

EXAMPLES:

sage: from itertools import cycle
sage: w = Word(cycle([9,8,4]))
sage: w
word: 9849849849849849849849849849849849849849...
sage: prefix = w[:23]
sage: prefix
word: 98498498498498498498498
sage: prefix.minimal_period()
3

>>> from sage.all import *
>>> from itertools import cycle
>>> w = Word(cycle([Integer(9),Integer(8),Integer(4)]))
>>> w
word: 9849849849849849849849849849849849849849...
>>> prefix = w[:Integer(23)]
>>> prefix
word: 98498498498498498498498
>>> prefix.minimal_period()
3
class sage.combinat.words.word.InfiniteWord_morphic(parent, morphism, letter, coding=None, length=+Infinity)[source]#

Bases: WordDatatype_morphic, InfiniteWord_class

Morphic word of infinite length.

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

Infinite words behave like a Python list : they can be sliced using square braquets to define for example a prefix or a factor.

EXAMPLES:

sage: m = WordMorphism('a->ab,b->a')
sage: w = m.fixed_point('a')
sage: w
word: abaababaabaababaababaabaababaabaababaaba...

>>> from sage.all import *
>>> m = WordMorphism('a->ab,b->a')
>>> w = m.fixed_point('a')
>>> w
word: abaababaabaababaababaabaababaabaababaaba...
sage.combinat.words.word.Word(data=None, alphabet=None, length=None, datatype=None, caching=True, RSK_data=None)[source]#

Construct a word.

INPUT:

  • data – (default: None) list, string, tuple, iterator, free monoid element, None (shorthand for []), or a callable defined on [0,1,...,length].

  • alphabet – any argument accepted by Words

  • length – (default: None) This is dependent on the type of data. It is ignored for words defined by lists, strings, tuples, etc., because they have a naturally defined length. For callables, this defines the domain of definition, which is assumed to be [0, 1, 2, ..., length-1]. For iterators: Infinity if you know the iterator will not terminate (default); "unknown" if you do not know whether the iterator terminates; "finite" if you know that the iterator terminates, but do not know the length.

  • datatype – (default: None) None, "list", "str", "tuple", "iter", "callable". If None, then the function tries to guess this from the data.

  • caching – (default: True) True or False. Whether to keep a cache of the letters computed by an iterator or callable.

  • RSK_data – (Optional. Default: None) A semistandard and a standard Young tableau to run the inverse RSK bijection on.

Note

Be careful when defining words using callables and iterators. It appears that islice does not pickle correctly causing various errors when reloading. Also, most iterators do not support copying and should not support pickling by extension.

EXAMPLES:

Empty word:

sage: Word()
word:

>>> from sage.all import *
>>> Word()
word:

Word with string:

sage: Word("abbabaab")
word: abbabaab

>>> from sage.all import *
>>> Word("abbabaab")
word: abbabaab

Word with string constructed from other types:

sage: Word([0,1,1,0,1,0,0,1], datatype="str")
word: 01101001
sage: Word((0,1,1,0,1,0,0,1), datatype="str")
word: 01101001

>>> from sage.all import *
>>> Word([Integer(0),Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0),Integer(1)], datatype="str")
word: 01101001
>>> Word((Integer(0),Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0),Integer(1)), datatype="str")
word: 01101001

Word with list:

sage: Word([0,1,1,0,1,0,0,1])
word: 01101001

>>> from sage.all import *
>>> Word([Integer(0),Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0),Integer(1)])
word: 01101001

Word with list constructed from other types:

sage: Word("01101001", datatype="list")
word: 01101001
sage: Word((0,1,1,0,1,0,0,1), datatype="list")
word: 01101001

>>> from sage.all import *
>>> Word("01101001", datatype="list")
word: 01101001
>>> Word((Integer(0),Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0),Integer(1)), datatype="list")
word: 01101001

Word with tuple:

sage: Word((0,1,1,0,1,0,0,1))
word: 01101001

>>> from sage.all import *
>>> Word((Integer(0),Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0),Integer(1)))
word: 01101001

Word with tuple constructed from other types:

sage: Word([0,1,1,0,1,0,0,1], datatype="tuple")
word: 01101001
sage: Word("01101001", datatype="str")
word: 01101001

>>> from sage.all import *
>>> Word([Integer(0),Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0),Integer(1)], datatype="tuple")
word: 01101001
>>> Word("01101001", datatype="str")
word: 01101001

Word with iterator:

sage: from itertools import count
sage: Word(count())
word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,...
sage: Word(iter("abbabaab")) # iterators default to infinite words
word: abbabaab
sage: Word(iter("abbabaab"), length="unknown")
word: abbabaab
sage: Word(iter("abbabaab"), length="finite")
word: abbabaab

>>> from sage.all import *
>>> from itertools import count
>>> Word(count())
word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,...
>>> Word(iter("abbabaab")) # iterators default to infinite words
word: abbabaab
>>> Word(iter("abbabaab"), length="unknown")
word: abbabaab
>>> Word(iter("abbabaab"), length="finite")
word: abbabaab

Word with function (a ‘callable’):

sage: f = lambda n : add(Integer(n).digits(2)) % 2
sage: Word(f)
word: 0110100110010110100101100110100110010110...
sage: Word(f, length=8)
word: 01101001

>>> from sage.all import *
>>> f = lambda n : add(Integer(n).digits(Integer(2))) % Integer(2)
>>> Word(f)
word: 0110100110010110100101100110100110010110...
>>> Word(f, length=Integer(8))
word: 01101001

Word over a string with a parent:

sage: w = Word("abbabaab", alphabet="abc"); w
word: abbabaab
sage: w.parent()
Finite words over {'a', 'b', 'c'}

>>> from sage.all import *
>>> w = Word("abbabaab", alphabet="abc"); w
word: abbabaab
>>> w.parent()
Finite words over {'a', 'b', 'c'}

Word from a free monoid element:

sage: M.<x,y,z> = FreeMonoid(3)
sage: Word(x^3*y*x*z^2*x)
word: xxxyxzzx

>>> from sage.all import *
>>> M = FreeMonoid(Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = M._first_ngens(3)
>>> Word(x**Integer(3)*y*x*z**Integer(2)*x)
word: xxxyxzzx

The default parent is the combinatorial class of all words:

sage: w = Word("abbabaab"); w
word: abbabaab
sage: w.parent()
Finite words over Set of Python objects of class 'object'

>>> from sage.all import *
>>> w = Word("abbabaab"); w
word: abbabaab
>>> w.parent()
Finite words over Set of Python objects of class 'object'

We can also input a semistandard tableau and a standard tableau to obtain a word from the inverse RSK algorithm using the RSK_data option:

sage: p = Tableau([[1,2,2],[3]]); q = Tableau([[1,2,4],[3]])
sage: Word(RSK_data=[p, q])
word: 1322

>>> from sage.all import *
>>> p = Tableau([[Integer(1),Integer(2),Integer(2)],[Integer(3)]]); q = Tableau([[Integer(1),Integer(2),Integer(4)],[Integer(3)]])
>>> Word(RSK_data=[p, q])
word: 1322
class sage.combinat.words.word.Word_iter(parent, iter, length=None)[source]#

Bases: WordDatatype_iter, Word_class

Word of unknown length (finite or infinite) represented by an iterable.

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

Words behave like a Python list : they can be sliced using square braquets to define for example a prefix or a factor.

EXAMPLES:

sage: w = Word(iter([1,1,4,9]*1000), length='unknown', caching=False)
sage: w
word: 1149114911491149114911491149114911491149...
sage: w.delta()
word: 2112112112112112112112112112112112112112...

>>> from sage.all import *
>>> w = Word(iter([Integer(1),Integer(1),Integer(4),Integer(9)]*Integer(1000)), length='unknown', caching=False)
>>> w
word: 1149114911491149114911491149114911491149...
>>> w.delta()
word: 2112112112112112112112112112112112112112...
class sage.combinat.words.word.Word_iter_with_caching(parent, iter, length=None)[source]#

Bases: WordDatatype_iter_with_caching, Word_class

Word of unknown length (finite or infinite) represented by an iterable (with caching).

For such word \(w\), type w. and hit Tab key to see the list of functions defined on \(w\).

Words behave like a Python list : they can be sliced using square braquets to define for example a prefix or a factor.

EXAMPLES:

sage: w = Word(iter([1,2,3]*1000), length='unknown')
sage: w
word: 1231231231231231231231231231231231231231...
sage: w.finite_differences(mod=2)
word: 1101101101101101101101101101101101101101...

>>> from sage.all import *
>>> w = Word(iter([Integer(1),Integer(2),Integer(3)]*Integer(1000)), length='unknown')
>>> w
word: 1231231231231231231231231231231231231231...
>>> w.finite_differences(mod=Integer(2))
word: 1101101101101101101101101101101101101101...