On the recognizability of self-generating sets.
The number of positions starting a square in binary words.
Compatibility relations on codes and free monoids
A compatibility relation on letters induces a reflexive and symmetric relation on words of equal length. We consider these word relations with respect to the theory of variable length codes and free monoids. We define an -code and an -free monoid for arbitrary word relations and . Modified Sardinas-Patterson algorithm is presented for testing whether finite sets of words are -codes. Coding capabilities of relational codes are measured algorithmically by finding minimal and maximal relations. We...
Transcendence of numbers with an expansion in a subclass of complexity 2 + 1
We divide infinite sequences of subword complexity into four subclasses with respect to left and right special elements and examine the structure of the subclasses with the help of Rauzy graphs. Let ≥ 2 be an integer. If the expansion in base of a number is an Arnoux-Rauzy word, then it belongs to Subclass I and the number is known to be transcendental. We prove the transcendence of numbers with expansions in the subclasses II and III.
The theorem of Fine and Wilf for relational periods
We consider relational periods, where the relation is a compatibility relation on words induced by a relation on letters. We prove a variant of the theorem of Fine and Wilf for a (pure) period and a relational period.
A Characterization of Multidimensional -Automatic Sequences
An infinite word is -automatic if, for all , its st letter is the output of a deterministic automaton fed with the representation of in the considered numeration system . In this extended abstract, we consider an analogous definition in a multidimensional setting and present the connection to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for , we state that a multidimensional infinite word over a finite alphabet is -automatic for some abstract numeration...
The theorem of Fine and Wilf for relational periods
We consider relational periods, where the relation is a compatibility relation on words induced by a relation on letters. We prove a variant of the theorem of Fine and Wilf for a (pure) period and a relational period.
On the number of squares in partial words
The theorem of Fraenkel and Simpson states that the maximum number of distinct squares that a word of length can contain is less than . This is based on the fact that no more than two squares can have their last occurrences starting at the same position. In this paper we show that the maximum number of the last occurrences of squares per position in a partial word containing one hole is , where is the size of the alphabet. Moreover, we prove that the number of distinct squares in a partial word...
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