Related papers: Convergence of Pascal-Like Triangles in Parry-Bert…
The probabilities for gaps in the eigenvalue spectrum of finite $ N\times N $ random unitary ensembles on the unit circle with a singular weight, and the related hermitian ensembles on the line with Cauchy weight, are found exactly. The…
In this paper, we investigate the existence of Sierpi\'{n}ski numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpi\'{n}ski numbers and Riesel numbers of the…
We consider a two-parameter family of triangles whose $(n,k)$-th entry (counting the initial entry as the $(0,0)$-th entry) is the number of tilings of $N$-boards (which are linear arrays of $N$ unit square cells for any nonnegative integer…
Many applications of denotational semantics, such as higher-order model checking or the complexity of normalization, rely on finite semantics for monomorphic type systems. We exhibit such a finite semantics for a polymorphic purely linear…
We study the convergence of resistance metrics and resistance forms on a converging sequence of spaces. As an application, we study the existence and uniqueness of self-similar Dirichlet forms on Sierpinski gaskets with added rotated…
Enumerating the number of times one word occurs in another is a much-studied combinatorial subject. By utilizing a method that we call ``lexicographic extreme referencing'', we provide a formula for computing occurrences of one binary word…
We introduce the notion of general prints of a word, which is substantialized by certain canonical decompositions, to study repetition in words. These associated decompositions, when applied recursively on a word, result in what we term as…
We generalize the concept of Pascal matrices to matrices associated with sets of points by considering multidimensional binomial coefficients as entries. We study their properties and prove that the infinite matrix associated with the set…
The notion of a two-dimensional word arises naturally in the study of combinatorics on words, while the iterative construction of pedal triangles results in a rich dynamical system in the study of geometry. At first, these two classes of…
We extend the classical Ostrowski numeration systems, closely related to Sturmian words, by allowing a wider range of coefficients, so that possible representations of a number $n$ better reflect the structure of the associated Sturmian…
Gaussian binomial coefficients are q-analogues of the binomial coefficients of integers. On the other hand, binomial coefficients have been extended to finite words, i.e., elements of the finitely generated free monoids. In this paper we…
The close relationship among the polynomial functions and Fibonacci numerical sequences is shown in this paper. These numerical sequences are defined by the recurrence equation $x_{k + n} = \displaystyle\sum_{j = 0}^{n-1}\alpha_j x_{k +…
In this paper, after presenting the results of the generalization of Pascal triangle (using powers of base numbers), we examine some properties of the 112-based triangle, most of all regarding to prime numbers. Additionally, an effective…
The numbers we study in this paper are of the form $B_{n, p}(k)$, which is the number of binary words of length $n$ that contain the word $p$ (as a subsequence) exactly $k$ times. Our motivation comes from the analogous study of pattern…
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…
The Pascal matrix, which is related to Pascal's triangle, appears in many places in the theory of uniform distribution and in many other areas of mathematics. Examples are the construction of low-discrepancy sequences as well as normal…
This paper studies the product $\bar{G}_n$ of the binomial coefficients in the n-th row of Pascal's triangle, which equals the reciprocal of the product of all the reduced and unreduced Farey fractions of order n. It studies its size as a…
We show that the convergence rate of the cusp winding spectrum to the Hausdorff dimension of the limit set of a generalized Schottky group with one parabolic generator is polynomial. Our main theorem provides the new phenomenon in which…
The trinomial transform of a sequence is a generalization of the well-known binomial transform, replacing binomial coefficients with trinomial coefficients. We examine Pascal-like triangles under trinomial transform, focusing on the ternary…
In this document we achieve exact and asymptotic enumeration of words, compositions over a finite group, and/or integer compositions characterized by local restrictions and, separately, subsequence pattern avoidance. We also count…