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    Generalized trapezoidal words

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    The factor complexity function Cw(n)C_w(n) of a finite or infinite word ww counts the number of distinct factors of ww of length nn for each n0n \ge 0. A finite word ww of length w|w| is said to be trapezoidal if the graph of its factor complexity Cw(n)C_w(n) as a function of nn (for 0nw0 \leq n \leq |w|) is that of a regular trapezoid (or possibly an isosceles triangle); that is, Cw(n)C_w(n) increases by 1 with each nn on some interval of length rr, then Cw(n)C_w(n) is constant on some interval of length ss, and finally Cw(n)C_w(n) decreases by 1 with each nn on an interval of the same length rr. Necessarily Cw(1)=2C_w(1)=2 (since there is one factor of length 00, namely the empty word), so any trapezoidal word is on a binary alphabet. Trapezoidal words were first introduced by de Luca (1999) when studying the behaviour of the factor complexity of finite Sturmian words, i.e., factors of infinite "cutting sequences", obtained by coding the sequence of cuts in an integer lattice over the positive quadrant of R2\mathbb{R}^2 made by a line of irrational slope. Every finite Sturmian word is trapezoidal, but not conversely. However, both families of words (trapezoidal and Sturmian) are special classes of so-called "rich words" (also known as "full words") - a wider family of finite and infinite words characterized by containing the maximal number of palindromes - studied in depth by the first author and others in 2009. In this paper, we introduce a natural generalization of trapezoidal words over an arbitrary finite alphabet A\mathcal{A}, called generalized trapezoidal words (or GT-words for short). In particular, we study combinatorial and structural properties of this new class of words, and we show that, unlike the binary case, not all GT-words are rich in palindromes when A3|\mathcal{A}| \geq 3, but we can describe all those that are rich.Comment: Major revisio

    Construction Of A Rich Word Containing Given Two Factors

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    A finite word ww with w=n\vert w\vert=n contains at most n+1n+1 distinct palindromic factors. If the bound n+1n+1 is attained, the word ww is called \emph{rich}. Let \Factor(w) be the set of factors of the word ww. It is known that there are pairs of rich words that cannot be factors of a common rich word. However it is an open question how to decide for a given pair of rich words u,vu,v if there is a rich word ww such that \{u,v\}\subseteq \Factor(w). We present a response to this open question:\\ If w1,w2,ww_1, w_2,w are rich words, m=max{w1,w2}m=\max{\{\vert w_1\vert,\vert w_2\vert\}}, and \{w_1,w_2\}\subseteq \Factor(w) then there exists also a rich word wˉ\bar w such that \{w_1,w_2\}\subseteq \Factor(\bar w) and wˉm2k(m)+2\vert \bar w\vert\leq m2^{k(m)+2}, where k(m)=(q+1)m2(4q10m)log2mk(m)=(q+1)m^2(4q^{10}m)^{\log_2{m}} and qq is the size of the alphabet. Hence it is enough to check all rich words of length equal or lower to m2k(m)+2m2^{k(m)+2} in order to decide if there is a rich word containing factors w1,w2w_1,w_2

    A Characterization of Infinite LSP Words

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    G. Fici proved that a finite word has a minimal suffix automaton if and only if all its left special factors occur as prefixes. He called LSP all finite and infinite words having this latter property. We characterize here infinite LSP words in terms of SS-adicity. More precisely we provide a finite set of morphisms SS and an automaton A{\cal A} such that an infinite word is LSP if and only if it is SS-adic and all its directive words are recognizable by A{\cal A}
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