اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدPattern 1^j0^i avoiding binary words
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تأليف
Stefano Bilotta
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In this paper we study the enumeration and the construction, according to the number of ones, of particular binary words avoiding a fixed pattern. The growth of such words can be described by particular jumping and marked succession rules. This approach enables us to obtain an algorithm which constructs all binary words having a fixed number of ones and then kills those containing the forbidden pattern.
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-maximal} if there does not exist another submonoid generated by at most $k$ words containing $M$. We call a set $X subseteq A^*$ {em primitive} if it is the basis of a $|X|$-maximal submonoid. This definition encompasses the notion of primitive word -- in fact, ${w}$ is a primitive set if and only if $w$ is a primitive word. By definition, for any set $X$, there exists a primitive set $Y$ such that $X subseteq Y^*$. We therefore call $Y$ a {em primitive root} of $X$. As a main result, we prove that if a set has rank $2$, then it has a unique primitive root. To obtain this result, we prove that the intersection of two $2$-maximal submonoids is either the empty word or a submonoid generated by one single primitive word. For a single word $w$, we say that the set ${x,y}$ is a {em bi-root} of $w$ if $w$ can be written as a concatenation of copies of $x$ and $y$ and ${x,y}$ is a primitive set. We prove that every primitive word $w$ has at most one bi-root ${x,y}$ such that $|x|+|y|<sqrt{|w|}$. That is, the bi-root of a word is unique provided the word is sufficiently long with respect to the size (sum of lengths) of the root. Our results are also compared to previous approaches that investigate pseudo-repetitions, where a morphic involutive function $theta$ is defined on $A^*$. In this setting, the notions of $theta$-power, $theta$-primitive and $theta$-root are defined, and it is shown that any word has a unique $theta$-primitive root. This result can be obtained with our approach by showing that a word $w$ is $theta$-primitive if and only if ${w, theta(w)}$ is a primitive set.
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