Related papers: On Number of Rich Words
Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many n, every length-n factor is a product of two palindromes. We show that every Sturmian word has this property,…
Lambda words are sequences obtained by encoding the differences between ordered elements of the form i+j\theta, where i and j are non-negative integers and 1 < \theta <2. Lambda words are right-infinite words defined over an infinite…
In this paper we explore various interconnections between rich words, Sturmian words, and trapezoidal words. Rich words, first introduced in arXiv:0801.1656 by the second and third authors together with J. Justin and S. Widmer, constitute a…
We investigate the number of sets of words that can be formed from a finite alphabet, counted by the total length of the words in the set. An explicit expression for the counting sequence is derived from the generating function, and…
In this paper we explore a new hierarchy of classes of languages and infinite words and its connection with complexity classes. Namely, we say that a language belongs to the class $L_k$ if it is a subset of the catenation of $k$ languages…
We investigate the problem of the maximum number of cubic subwords (of the form $www$) in a given word. We also consider square subwords (of the form $ww$). The problem of the maximum number of squares in a word is not well understood.…
Free words are elements of a free monoid, generated over an alphabet via the binary operation of concatenation. Casually speaking, a free word is a finite string of letters. Henceforth, we simply refer to them as words. Motivated by recent…
Recently the Fibonacci word $W$ on an infinite alphabet was introduced by [Zhang et al., Electronic J. Combinatorics 24-2 (2017) #P2.52] as a fixed point of the morphism $\phi: (2i) \mapsto (2i)(2i+ 1),\ (2i+ 1) \mapsto (2i+ 2)$ over all $i…
Let $\mathfrak A$ be an alphabet and $W$ be a set of words in the free monoid ${\mathfrak A}^*$. Let $S(W)$ denote the Rees quotient over the ideal of ${\mathfrak A}^*$ consisting of all words that are not subwords of words in $W$. A set of…
A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern $p$ is unavoidable if, over every finite alphabet, every sufficiently long word encounters $p$. A theorem by…
The prefix palindromic length $p_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. This function is surprisingly difficult to…
We focus on infinite words with languages closed under reversal. If frequencies of all factors are well defined, we show that the number of different frequencies of factors of length n+1 does not exceed 2C(n+1)-2C(n)+1.
A word $w_1w_2\cdots w_n$ is said to be up-down if $w_1 < w_2 >w_3 \cdots$. Carlitz and Scoville found the generating function for the number of up-down words over an alphabet of size $k$. Using properties of the Chebyshev polynomials we…
If w is a word in d>1 letters and G is a finite group, evaluation of w on a uniformly randomly chosen d-tuple in G gives a random variable with values in G, which may or may not be uniform. It is known that if G ranges over finite simple…
We focus on $\Theta$-rich and almost $\Theta$-rich words over a finite alphabet $\mathcal{A}$, where $\Theta$ is an involutive antimorphism over $\mathcal{A}^*$. We show that any recurrent almost $\Theta$-rich word $\uu$ is an image of a…
A finite set S of words over the alphabet A is called non-complete if Fact(S*) is different from A*. A word w in A* - Fact(S*) is said to be uncompletable. We present a series of non-complete sets S_k whose minimal uncompletable words have…
Ulam words are binary words defined recursively as follows: the length-$1$ Ulam words are $0$ and $1$, and a binary word of length $n$ is Ulam if and only if it is expressible uniquely as a concatenation of two shorter, distinct Ulam words.…
We find the lexicographically least infinite binary rich word having critical exponent $2+\sqrt{2}/2$
The subword complexity of a word $w$ over a finite alphabet $\mathcal{A}$ is a function that assigns for each positive integer $n$, the number of distinct subwords of length $n$ in $w$. The subword complexity of a word is a good measure of…
The repetition threshold for words on $n$ letters, denoted $\mbox{RT}(n)$, is the infimum of the set of all $r$ such that there are arbitrarily long $r$-free words over $n$ letters. A repetition threshold for circular words on $n$ letters…