Related papers: Palindromic Density
A natural number N is said to be palindromic if its binary representation reads the same forwards and backwards. In this paper we study the quotients of two palindromic numbers and answer some basic questions about the resulting sets of…
This paper studies the density of zero and one in subwords of the Fibonacci word with lengths less than thirty and compares them to the densities of their corresponding palindromes. We used computational methods to produce a sufficiently…
We investigate the least number of palindromic factors in an infinite word. We first consider general alphabets, and give answers to this problem for periodic and non-periodic words, closed or not under reversal of factors. We then…
A palindromic periodicity is a factor of an infinite word $(ps)^\omega$ where $p$ and $s$ are palindromes and the factor has length at least $|ps|$, for example, $accabaccab$. In this paper we describe several ways in which a palindromic…
We prove a precise formula for the minimal number K(n) such that every binary word of length $n$ can be divided into K(n) palindromes. Also we estimate the average number $\ol K(n)$ of palindromes composing a random binary word of the…
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…
In this article we generalize packing density problems from permutations to patterns with repeated letters and generalized patterns. We are able to find the packing density for some classes of patterns and several other short patterns.
Imagine a sequence in which the first letter comes from a binary alphabet, the second letter can be chosen on an alphabet with 10 elements, the third letter can be chosen on an alphabet with 3 elements and so on. When such a sequence can be…
A palindrome is a word that reads the same left-to-right as right-to-left. We show that every simple group has a finite generating set $X$, such that every element of it can be written as a palindrome in the letters of $X$. Moreover, every…
We introduce a notion of palindromicity of a natural number which is independent of the base. We study the existence and density of palindromic and multiple palindromic numbers, and we raise several related questions.
Two results on palindromicity of bi-infinite words in a finite alphabet are presented. The first is a simple, but efficient criterion to exclude palindromicity of minimal sequences and applies, in particular, to the Rudin-Shapiro sequence.…
Let $k \ge 2$ and consider the sequence $\{P_n^{(k)}\}_{n \ge 2-k}$ of $k$-generalized Pell numbers, which begins with the first $k$ terms as $0, \ldots, 0, 0, 1$, and satisfies the recurrence relation $P_n^{(k)} = 2P_{n-1}^{(k)} +…
We present a method which displays all palindromes of a given length from De Bruijn words of a certain order, and also a recursive one which constructs all palindromes of length $n+1$ from the set of palindromes of length $n$. We show that…
We study the distribution of palindromic numbers (with respect to a fixed base $g\ge 2$) over certain congruence classes, and we derive a nontrivial upper bound for the number of prime palindromes $n\le x$ as $x\to\infty$. Our results show…
In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism…
We study the palindromic complexity of infinite words $u_\beta$, the fixed points of the substitution over a binary alphabet, $\phi(0)=0^a1$, $\phi(1)=0^b1$, with $a-1\geq b\geq 1$, which are canonically associated with quadratic non-simple…
Recently, a new characterization of Lyndon words that are also perfectly clustering was proposed by Lapointe and Reutenauer (2024). A word over a ternary alphabet {a,b,c} is called perfectly clustering Lyndon if and only if it is the…
In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the…
We introduce and study a subclass of joint Bernoulli distributions which has the palindromic property. For such distributions the vector of joint probabilities is unchanged when the order of the elements is reversed. We prove for binary…
We introduce two new classes of integers. The first class consists of numbers $N$ for which there exists at least one nonnegative integer $A$, such that the sum of $A$ and the sum of digits of $N$, added to the reversal of the sum, gives…