Word classes#
AUTHORS:
Arnaud Bergeron
Amy Glen
Sébastien Labbé
Franco Saliola
- class sage.combinat.words.word.FiniteWord_callable(parent, callable, length=None)#
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
- class sage.combinat.words.word.FiniteWord_callable_with_caching(parent, callable, length=None)#
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
- class sage.combinat.words.word.FiniteWord_char#
Bases:
WordDatatype_char
,FiniteWord_class
Finite word represented by an array
unsigned char *
(i.e. integers between 0 and 255).For any word
w
, typew.<TAB>
to see the functions that can be applied tow
.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) # optional - 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
- class sage.combinat.words.word.FiniteWord_iter(parent, iter, length=None)#
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
- class sage.combinat.words.word.FiniteWord_iter_with_caching(parent, iter, length=None)#
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
- class sage.combinat.words.word.FiniteWord_list#
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...
- class sage.combinat.words.word.FiniteWord_morphic(parent, morphism, letter, coding=None, length=+Infinity)#
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
- class sage.combinat.words.word.FiniteWord_str#
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
- class sage.combinat.words.word.FiniteWord_tuple#
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
- class sage.combinat.words.word.InfiniteWord_callable(parent, callable, length=None)#
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...
- class sage.combinat.words.word.InfiniteWord_callable_with_caching(parent, callable, length=None)#
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
- class sage.combinat.words.word.InfiniteWord_iter(parent, iter, length=None)#
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
- class sage.combinat.words.word.InfiniteWord_iter_with_caching(parent, iter, length=None)#
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
- class sage.combinat.words.word.InfiniteWord_morphic(parent, morphism, letter, coding=None, length=+Infinity)#
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...
- sage.combinat.words.word.Word(data=None, alphabet=None, length=None, datatype=None, caching=True, RSK_data=None)#
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 Wordslength
– (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"
. IfNone
, then the function tries to guess this from the data.caching
– (default:True
)True
orFalse
. 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:
Word with string:
sage: 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
Word with list:
sage: Word([0,1,1,0,1,0,0,1]) 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
Word with tuple:
sage: Word((0,1,1,0,1,0,0,1)) 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
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
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
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'}
Word from a free monoid element:
sage: M.<x,y,z> = FreeMonoid(3) sage: Word(x^3*y*x*z^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'
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
- class sage.combinat.words.word.Word_iter(parent, iter, length=None)#
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...
- class sage.combinat.words.word.Word_iter_with_caching(parent, iter, length=None)#
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...