Generalized Pascal Triangles Research Articles

MORE ON THE GENERALIZED PASCAL TRIANGLES

The aim of this article is to obtain a new factorization of a generalized Pascal triangle. This factorization particularly emphasizes that there is a close relation between generalized Pascal triangles and Toeplitz matrices. However, we will show that in general there is no such relation between generalized Pascal triangles and Hankel matrices.

Symmetric binary Steinhaus triangles and parity-regular Steinhaus graphs

A binary Steinhaus triangle is a triangle of zeroes and ones that points down and with the same local rule as the Pascal triangle modulo 2. A binary Steinhaus triangle is said to be rotationally symmetric, horizontally symmetric or dihedrally symmetric if it is invariant under the 120 degrees rotation, the horizontal reflection or both, respectively. The first part of this paper is devoted to the study of linear subspaces of rotationally symmetric, horizontally symmetric and dihedrally symmetric binary Steinhaus triangles. We obtain simple explicit bases for each of them by using elementary properties of the binomial coefficients. A Steinhaus graph is a simple graph with an adjacency matrix whose upper-triangular part is a binary Steinhaus triangle. A Steinhaus graph is said to be even or odd if all its vertex degrees are even or odd, respectively. One of the main results of this paper is the existence of an isomorphism between the linear subspace of even Steinhaus graphs and a certain linear subspace of dihedrally symmetric binary Steinhaus triangles. This permits us to give, in the second part of this paper, an explicit basis for even Steinhaus graphs and for the vector space of parity-regular Steinhaus graphs; i.e., the linear subspace of Steinhaus graphs that are even or odd. Finally, in the last part of this paper, we consider the generalized Pascal triangles, that are triangles of zeroes and ones, that point up now, and always with the same local rule as the Pascal triangle modulo 2. New simple bases for each linear subspace of symmetric generalized Pascal triangles are deduced from the results of the first part.

Q-total positivity and strong q-log-convexity for some generalized triangular arrays

The aim of this paper is twofold. First, it proves the total positivity of the generalized Pascal triangle (classical and q-analogue versions). Second, it studies of the log-convexity (resp. strong q-log-convexity) of the first column of certain infinite generalized triangular array $$\{A_{n,k}\}_{0\le k\le 2n}$$ of nonnegative numbers (resp. of polynomials in q with nonnegative coefficients), of the form $$A_{n,k}=e_{k}A_{n-1,k-2}+f_{k}A_{n-1,k-1}+g_{k}A_{n-1,k}+h_{k}A_{n-1,k+1}+i_{k}A_{n-1,k+2}$$ . This allows a unified treatment of the log-convexity of the peers sequence of central trinomials $$\left\{ \left( {\begin{array}{c}2n\\ 2n\end{array}}\right) _{2}\right\} _{n\ge 0}$$ , as well as that of the strong q-log-convexity of the polynomial of square trinomial coefficients, i.e. $$A_{n}(q)=\sum _{k=0}^{2n}\left( {\begin{array}{c}n\\ k\end{array}}\right) _{2}^{2}q^{k}$$ .

Computing Sticks against Random Walk

A new deterministic model with the help of geometric constructions and computing sticks (not related to trajectories) is proposed for the new justification of consistency of the probabilistic approach to explain the random walk on a plane. A new, stepped form of the arithmetic triangle of Pascal based on the construction of horizontal and vertical lines (arrows) is suggested, a comparison is made with Pascal’s triangle of the usual form. A two-sided generalization of Pascal’s triangle is proposed. Geometric constructions and formulas for calculating the coefficients that fill in these new geometric (arithmetic) figures are given. Further types of generalization of the step-shaped Pascal triangle are proposed. Examples of generalized initial conditions and generalized recursive formulas for constructing various types of a generalized Pascal triangle are given.

Garside combinatorics for Thompson’s monoid $F^+$ and a hybrid with the braid monoid $B_{\infty }^{+}$

On the model of simple braids, defined to be the left divisors of Garside's elements $\Delta_n$ in the monoid $B_\infty^+$ , we investigate simple elements in Thompson's monoid $F^+$ and in a larger monoid $H^+$ that is a hybrid of $B_\infty^+$ and $F^+$ : in both cases, we count how many simple elements left divide the right lcm of the first n − 1 atoms, and characterize their normal forms in terms of forbidden factors. In the case of $H^+$, a generalized Pascal triangle appears.

Application of generalized pascal triangle for description of oscillations of friction forces

This article describes the application of the generalized Pascal triangle for description of fixed variation of the oscillation amplitude of friction force upon frictional interaction of materials, and the region of interest in terms of friction theory and material wear is highlighted. Mathematical models of friction are presented, and their efficiency is confirmed by experimental data.

Behavior of Digital Sequences Through Exotic Numeration Systems

Many digital functions studied in the literature, e.g., the summatory function of the base-$k$ sum-of-digits function, have a behavior showing some periodic fluctuation. Such functions are usually studied using techniques from analytic number theory or linear algebra. In this paper we develop a method based on exotic numeration systems and we apply it on two examples motivated by the study of generalized Pascal triangles and binomial coefficients of words.

Counting the number of non-zero coefficients in rows of generalized Pascal triangles

This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence (S(n))n≥0 counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for (S(n))n≥0. This leads to a connection with the 2-regular Stern–Brocot sequence and the sequence of denominators occurring in the Farey tree. Then we extend our construction to the Zeckendorf numeration system based on the Fibonacci sequence. Again our tree structure permits us to obtain recurrence relations for and the F-regularity of the corresponding sequence.

Preserving Log-Convexity for Generalized Pascal Triangles

We establish the preserving log-convexity property for the generalized Pascal triangles. It is an extension of a result of H. Davenport and G. Pólya "On the product of two power series", who proved that the binomial convolution of two log-convex sequences is log-convex.

Symmetries in Steinhaus Triangles and in Generalized Pascal Triangles

AbstractWe give explicit formulae for obtaining the binary sequences which produce Steinhaus triangles and generalized Pascal triangles with rotational and dihedral symmetries.

New families of integer matrices whose leading principal minors form some well-known sequences

The purpose of this article is to obtain some new infinite families of Toeplitz matri- ces, 7-matrices and generalized Pascal triangles whose leading principal minors form the Fibonacci, Lucas, Pell and Jacobsthal sequences. We also present a new proof for Theorem 3.1 in (R. Bacher. Determinants of matrices related to the Pascal triangle. J. Theor. Nombres Bordeaux, 14:19-41, 2002.).

A universal sequence of integers generating balanced Steinhaus figures modulo an odd number

In this paper, we partially solve an open problem, due to J.C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer n, that are Steinhaus triangles containing all the elements of Z / n Z with the same multiplicity. For every odd number n, we build an orbit in Z / n Z , by the linear cellular automaton generating the Pascal triangle modulo n, which contains infinitely many balanced Steinhaus triangles. This orbit, in Z / n Z , is obtained from an integer sequence called the universal sequence. We show that there exist balanced Steinhaus triangles for at least 2/3 of the admissible sizes, in the case where n is an odd prime power. Other balanced Steinhaus figures, such as Steinhaus trapezoids, generalized Pascal triangles, Pascal trapezoids or lozenges, also appear in the orbit of the universal sequence modulo n odd. We prove the existence of balanced generalized Pascal triangles for at least 2/3 of the admissible sizes, in the case where n is an odd prime power, and the existence of balanced lozenges for all admissible sizes, in the case where n is a square-free odd number.

The Influence of the A-Current on the Dynamics of an Oscillator-Follower Inhibitory Network

The transient potassium A-current is present in almost all neurons and plays an essential role in determining the timing and frequency of action potential generation. We use a three-variable mathematical model to examine the role of the A-current in a rhythmic inhibitory network, as is common in central pattern generation. We focus on a feed-forward architecture consisting of an oscillator neuron inhibiting a follower neuron. We use separation of time scales to demonstrate that the trajectory of the follower neuron within each cycle can be tracked by analyzing the dynamics on a 2-dimensional slow manifold that as determined by the two slow model variables: the recovery variable and the inactivation of the A-current. The steady-state trajectory, however, requires tracking the slow variables across multiple cycles. We show that tracking the slow variables, under simplifying assumptions, leads to a one-dimensional map of the unit interval with at most a single discontinuity depending on g(A), the maximal conductance of the A-current, or other model parameters. We demonstrate that, as the value of g(A) is varied, the trajectory of the follower neuron goes through a set of bifurcations to produce n:m periodic solutions where the follower neuron becomes active m times for each n cycles of the oscillator. Using a generalized Pascal triangle, each n:m trajectory can be constructed as a combination of solutions from a higher level of the triangle.

Certain matrices related to Fibonacci sequence having recursive entries

Let φ = (φi)i≥1 and ψ = (ψi)i≥1 be two arbitrary sequences with φ1 = ψ1. LetAφ,ψ(n) denote the matrix of order n with entries ai,j , 1 ≤ i, j ≤ n, where a1,j = φj and ai,1 = ψi for 1 ≤ i ≤ n, and where ai,j = ai−1,j−1 + ai−1,j for 2 ≤ i, j ≤ n. It is of interest to evaluate the determinant of Aφ,ψ(n), where one of the sequences φ or ψ is the Fibonacci sequence (i.e., 1, 1, 2, 3, 5, 8,...) and the other is one of the following sequences: {k-times}α(k) = (1, 1,..., 1, 0, 0, 0,...) ,χ(k) = (1k, 2k, 3k,...,ik,...),ξ(k) = (1, k, k2,...,ki−1,...) (a geometric sequence),γ(k) = (1, 1 + k, 1+2k, ... , 1+(i − 1)k, .. .) (an arithmetic sequence). For some sequences of the above type the inverse of Aφ,ψ(n) is found. In the final part of this paper, the determinant of a generalized Pascal triangle associated to the Fibonacci sequence is found.

Restricted Occupancy of S Kinds of Cells And Generalized Pascal Triangles

Restricted Occupancy of S Kinds of Cells And Generalized Pascal Triangles

Theories of generalized Pascal triangles

Theories of generalized Pascal triangles

Generalized Pascal Triangles and Pyramids

Generalized Pascal Triangles and Pyramids

Book Announcement: “Generalized Pascal Triangles and Pyramids”

Book Announcement: “Generalized Pascal Triangles and Pyramids”

Irrational speeds of configurations growth in generalized Pascal triangles

A method is presented for obtaining irrational speeds of growth of configurations in generalized Pascal triangles and one-dimensional cellular automata. This method does not use diagonalization or simulation of Turing machines but uses a number-theoretic idea.

Generating Functions of Central Values in Generalized Pascal Triangles

Generating Functions of Central Values in Generalized Pascal Triangles