Related papers: Balanced Fibonacci word rectangles, and beyond
There exists an exact relationship between the quasi-exactly solvable problems of quantum mechanics and models of square and rectangular random complex matrices. This relationship enables one to reduce the problem of constructing…
We investigate the scattered palindromic subwords in a finite word. We start by characterizing the words with the least number of scattered palindromic subwords. Then, we give an upper bound for the total number of palindromic subwords in a…
In this paper we completely solve the family of parametrised Thue equations \[ X(X-F_n Y)(X-2^n Y)-Y^3=\pm 1, \] where $F_n$ is the $n$-th Fibonacci number. In particular, for any integer $n\geq 3$ the Thue equation has only the trivial…
Minimal forbidden factors are a useful tool for investigating properties of words and languages. Two factorial languages are distinct if and only if they have different (antifactorial) sets of minimal forbidden factors. There exist…
We exhibit the construction of a deterministic automaton that, given k > 0, recognizes the (regular) language of k-differentiable words. Our approach follows a scheme of Crochemore et al. based on minimal forbidden words. We extend this…
We study the abelian period sets of Sturmian words, which are codings of irrational rotations on a one-dimensional torus. The main result states that the minimum abelian period of a factor of a Sturmian word of angle $\alpha$ with continued…
Two finite words are k-binomially equivalent if each subword (i.e., subsequence) of length at most k occurs the same number of times in both words. The k-binomial complexity of an infinite word is a function that maps the integer $n\geq 0$…
A $4^-$-power is a non-empty word of the form $XXXX^-$, where $X^-$ is obtained from $X$ by erasing the last letter. A binary word is called {\em faux-bonacci} if it contains no $4^-$-powers, and no factor 11. We show that faux-bonacci…
We show that a special case of the Feferman-Vaught composition theorem gives rise to a natural notion of automata for finite words over an infinite alphabet, with good closure and decidability properties, as well as several logical…
The Fibonacci word is the fixed point beginning with the letter $a$ of morphism $\sigma(a)=ab$, $\sigma(b)=a$ defined over the alphabet $\{a,b\}$. In this paper, we get explicit expression of the number of distinct fractional powers in each…
Let n be an even positive integer and F be the field \GF(2). A word in F^n is called balanced if its Hamming weight is n/2. A subset C \subseteq F^n$ is called a balancing set if for every word y \in F^n there is a word x \in C such that y…
In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a "wide sense"…
Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a word composed of $k$ pairwise distinct, concatenated words of equal length. Berger and Defant conjecture that for any sufficiently well-behaved aperiodic morphic word $w$,…
In previous papers, for an arithmetical function $f_0$, we defined functions $f_m$ and $c_m$ and designated numbers of restricted words over a finite alphabet counted by these functions. In this paper, we examine the reverse problem for…
The property of balance (in the sense of Feder and Mihail) is investigated in the context of paving matroids. The following examples are exhibited: (a) a class of ``sparse'' paving matroids that are balanced, but at the same time rich…
The hyperbolic Pascal triangle ${\cal HPT}_{4,q}$ $(q\ge5)$ is a new mathematical construction, which is a geometrical generalization of Pascal's arithmetical triangle. In the present study we show that a natural pattern of rows of ${\cal…
We consider the tiling of an $n$-board (a $1\times n$ array of square cells of unit width) with half-squares ($\frac12\times1$ tiles) and $(\frac12,\frac12)$-fence tiles. A $(\frac12,\frac12)$-fence tile is composed of two half-squares…
We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for $\pi$ involving Fibonacci and Lucas numbers. Our findings complement those…
Timed words are words where letters of the alphabet come with time stamps. We extend the definitions of semistandard tableaux, insertion, Knuth equivalence, and the plactic monoid to the setting of timed words. Using this, Greene's theorem…
We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer…