Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We also propose two more general conjectures.
We give an overview about some elementary properties of Hoggatt matrices, which are generalizations of Pascal triangle, and study q-analogs and Fibonacci analogs and derive a common generalization.
We study the Hankel transforms of sequences related to the central coefficients of a family of Pascal-like triangles. The mechanism of Riordan arrays is used to elucidate the structure of these transforms.
In this paper, firstly, by a determinant of deformed Pascal's triangle, namely the normalized Hessenberg matrix determinant, to count Dyck paths, we give another combinatorial proof of the theorems which are of Catalan numbers determinant representations and the recurrence formula. Secondly, a determinant of normalized Toeplitz-Hessenberg matrix, whose entries are binomials, arising in power series, we derive new four properties of Pascal's triangle.
Using Chebyshev polynomialsof both kinds, we construct rational fractions which are convergents of the smallest root of $x^2-αx+1$ for $α=3,4,5,\dots$.Some of the underlying identities suggest an identity involving binomialcoefficients which leads to a triangular array sharing many propertieswith Pascal's triangle.
This note collects some facts and conjectures about the Hankel determinants and their generating functions of the columns of Hoggatt triangles which apparently are related to combinatorial objects such as Young tableaux and Narayana numbers.
We define and characterize the $γ$-matrix associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. We also define and characterize the $γ$-matrix of the reversions of these triangles, in the case of ordinary Riordan arrays. We are led to the $γ$-matrices of a one-parameter family of generalized Narayana triangles. Thus these matrices generalize the matrix of $γ$-vectors of the associahedron. The principal tools used are the bivariate generating functions of the triangles and Jacobi continued fractions.
The following work shows new connections between the constants $π$ and $e$ with Pascal's triangle and the Lucas triangle, established via Fibonacci polynomials and similar means. Furthermore, relations between the two famous constants and the rows of Pascal's triangle and the Lucas triangle are conjectured, together with some other important related identities.
We pursue the investigation of generalizations of the Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. The finite words occurring in this paper belong to the language of a Parry numeration system satisfying the Bertrand property, i.e., we can add or remove trailing zeroes to valid representations. It is a folklore fact that the Sierpiński gasket is the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from the classical …
A "truncation" of Pascal's triangle is a triangular array of numbers that satisfies the usual Pascal recurrence but with a boundary condition that declares some terminal set of numbers along each row of the array to be zero. Presented here is a family of natural truncations of Pascal's triangle that generalize a kind of Catalan triangle. The numbers in each array are realized as differences of binomial coefficients, as counts of certain lattice paths and tableaux, and as entries of representing matrices for certain linear transformations of polynomial spaces. Lucas's theorem is applied to dete…
Menon's proof of the preservation of log-concavity of sequences under convolution becomes simpler when adapted to 2-sided infinite sequences. Under assumption of log-concavity of two 2-sided infinite sequences, the existence of the convolution is characterised by a convergence criterion. Preservation of log-concavity under convolution yields a method of establishing the log-concavity of rows of a large class of Pascal type triangles, including a weighted generalization of the Delannoy triangle. This method is also compared with known techniques of proving log-concavity.
We define and characterize the $f$-matrices associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. These generalize the face matrices of simplices and hypercubes. Their generating functions can be expressed simply in terms of continued fractions, which are shown to be transformations of the generating functions of the corresponding $γ$- and $h$-matrices.
We continue to investigate combinatorial properties of functions $f_m$ and $c_m$ considered in our previous papers. They depend on an initial arithmetic function $f_0$. In this paper, the values of $f_0$ are the binomial coefficients. We first consider the case when the values of $f_0$ are the binomial coefficients from a row of the Pascal triangle. The values of $f_0$ consider next are the binomial coefficients from a diagonal of the Pascal triangle. In two final cases, the values of $f_0$ are the central binomial coefficients and its adjacent neighbors. In each case, we derive an explicit fo…
We consider two families of Pascal-like triangles that have all ones on the left side and ones separated by $m-1$ zeros on the right side. The $m=1$ cases are Pascal's triangle and the two families also coincide when $m=2$. Members of the first family obey Pascal's recurrence everywhere inside the triangle. We show that the $m$-th triangle can also be obtained by reversing the elements up to and including the main diagonal in each row of the $(1/(1-x^m),x/(1-x))$ Riordan array. Properties of this family of triangles can be obtained quickly as a result. The $(n,k)$-th entry in the $m$-th member…
A new generalization of Pascal's triangle, the so-called hyperbolic Pascal triangles were introduced in [H.B, L.N, L.Sz: Hyperbolic Pascal triangles]. The mathematical background goes back to the regular mosaics in the hyperbolic plane. The alternating sum of elements in the rows was given in the special case $\{4,5\}$ of the hyperbolic Pascal triangles. In this article, we determine the alternating sum generally in the hyperbolic Pascal triangle corresponding to $\{4,q\}$ with $q\ge5$.
An elegant procedure which characterizes a decomposition of some class of binomial configurations into two other, resembling a definition of Pascal's Triangle, was given in \cite{gevay}. In essence, this construction was already presented in \cite{perspect}. We show that such a procedure is a result of fixing in configurations in some class $\mathcal K$ suitable hyperplanes which both: are in this class, and deleting such a hyperplane results in a configuration in this class. By a way of example we show two more (added to that of \cite{gevay}) natural classes of such configurations, discuss so…
We discuss the prominent role played by bilateral symmetry and modified Pascal triangles in self twin games, a subset of constant sum homogeneous weighted majority games. We show that bilateral symmetry of the free representations unequivocally identifies and characterizes this class of games and that modified Pascal triangles describe their cardinality for combinations of m and k, respectively linked through linear transforms to the key parameters n, number of players and h, number of types in the game. Besides, we derive the whole set of self twin games in the form of a genealogical tree obt…
Random walks are a series of up, down, and level steps that enumerate distinct paths from $(0,0)$ to $(2n,0)$, where $n$ is the semi-length of the path. We used these paths to analyze Catalan, Schröder, and Motzkin number sequences through a combination of matrix operations, quadratic functions, and inductive reasoning. Our results revealed a number of distinct patterns, some unnamed, between these number sequences and Pascal's triangle that can be explained through generating functions, first returns, group theory, and the Riordan matrix. Various proofs and properties of these number sequence…
We derive some, seemingly new, curious additive relations in the Pascal triangle. They arise in summing up the numbers in the triangle along some vertical line up to some place.
The classic way to write down Pascal's triangle leads to entries in alternating rows being vertically aligned. In this paper, we prove a linear dependence on vertically aligned entries in Pascal's triangle. Furthermore, we give an application of this dependence to morphisms between hyperelliptic curves.
Recently, a new generalization of Pascal's triangle, the so-called hyperbolic Pascal triangles were introduced. The mathematical background goes back to the regular mosaics in the hyperbolic plane. In this article, we investigate the paths in the hyperbolic Pascal triangle corresponding to the regular mosaic $\{4,5\}$, in which the binary recursive sequences $f_{n}=αf_{n-1}\pm f_{n-2}$ are represented ($α\in\mathbb{N}^+$).
We give a generalization of the Pascal triangle called the quasi s-Pascal triangle where the sum of the elements crossing the diagonal rays produce the s-bonacci sequence. For this, consider a lattice path in the plane whose step set is {L = (1, 0), L1 = (1, 1), L2 = (2, 1), . . . , Ls = (s, 1)}; an explicit formula is given. Thereby linking the elements of the quasi s-Pascal triangle with the bisnomial coefficients. We establish the recurrence relation for the sum of elements lying over any finite ray of the quasi s-Pascal triangle. The generating function of the cited sums is produced. We al…
In this paper, we introduce a new generalization of Pascal's triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We describe precisely the procedure of how to obtain a given type of hyperbolic Pascal triangle from a mosaic. Then we study certain quantitative properties such as the number, the sum, and the alternating sum of the elements of a row. Moreover, the pattern of the rows, and the appearence of some binary recurrences in a fixed hyperbolic triangle are investigated.
We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpiński gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo $2$, we describe and study the first properties of the subset of $[0, 1] \times [0, 1]$ associated with this extended Pascal triangle modulo a prime $p$.
We show that the $n$'th diagonal sum of Barry's modified Pascal triangle can be described as the continuant of the run lengths of the binary representation of $n$. We also obtain an explicit description for the row sums.
In this note we prove an assertion made by M. Levin in 1999: the Pascal matrix modulo 2 has the property that each of the square sub-matrices laying on the upper border or on the left border has determinants, computed in $\mathbb{Z}$, equal to 1 or -1.
In this paper, we describe a method to determine the power sum of the elements of the rows in the hyperbolic Pascal triangles corresponding to $\{4,q\}$ with $q\ge5$. The method is based on the theory of linear recurrences, and the results are demonstrated by evaluating the $k^{th}$ power sum in the range $2\le k\le 11$.
We find the eigenvalues and eigenvectors of the n by n matrix with (i,j) entry \binom(i-1,n-j), establishing a conjecture of Peele and Stanica. Curiously, the eigenvectors can be chosen to form a matrix which is its own inverse.
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 ${\cal HPT}_{4,5}$ is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the hyperbolic Pascal triangles. The geometrical properties of a ${\cal HPT}_{4,q}$ imply a graph structure between the finite Fibonacci words.
We use some basic properties of binomial and Stirling numbers to prove that the Euler characteristic is, essentially, the unique numerical topological invariant for compact polyhedra which can be expressed as a linear combination of the numbers of faces of triangulations. We obtain this result converting it into an eigenvalue problem.
The nth row of Pascal's trinomial triangle gives coefficients of (1+x+x^2)^n. Let g(n) denote the number of such coefficients that are odd. We review Moshe's algorithm for evaluating asymptotics of g(n) -- this involves computing the Lyapunov exponent for certain 2x2 random matrix products -- and then analyze further examples with more terms and higher powers of x.
The Pascal rhombus is a variant of Pascal's triangle in which each term is a sum of four earlier terms. Klostermeyer et al. made four conjectures about the Pascal rhombus modulo 2. In this paper we show how exploration of the stealth shape leads to unified proofs of all of these conjectures.
If L, respectively R are matrices with entries binom{i-1,j-1}, respectively binom{i-1,n-j}, it is known that L^2 = I (mod 2), respectively R^3 = I (mod 2), where I is the identity matrix of dimension n > 1 (see P10735-May 1999 issue of the American Mathematical Monthly). We generalize it for any prime p, and give a beautiful connection to Fibonacci numbers.
We study the combinatorial representation theory of the ``planar rook algebra" $P_n$. This algebra has a basis consisting of planar rook diagrams and multiplication given by diagram concatenation. For each integer $0 \le k \le n$, we construct natural representations $V^n_k$ which form a complete set of non-isomorphic, irreducible $P_n$-representations. We explicitly decompose the regular representation of $P_n$ into a direct sum of irreducible modules. We compute the Bratteli diagram for the tower of algebras $P_0 \subseteq P_1 \subseteq P_2 \subseteq ...$ and show that this Bratteli diagram …
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 $(S(n))_{n\ge 0}$ counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for $(S(n))_{n\ge 0}$. 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…
We prove several evaluations of determinants of matrices, the entries of which are given by the recurrence $a_{i,j}=a_{i-1,j}+a_{i,j-1}$, or variations thereof. These evaluations were either conjectured or extend conjectures by Roland Bacher [J. Théorie Nombres Bordeaux 13 (2001), to appear; see http://arxiv.org/abs/math.CO/0109013].
In this article we introduce a new geometric object called hyperbolic Pascal simplex. This new object is presented by the regular hypercube mosaic in the 4-dimensional hyperbolic space. The definition of the hyperbolic Pascal simplex, whose hyperfaces are hyperbolic Pascal pyramids and faces are hyperbolic Pascals triangles, is a natural generalization of the definition of the hyperbolic Pascal triangle and pyramid. We describe the growing of the hyperbolic Pascal simplex considering the numbers and the values of the elements. Further figures illustrate the stepping from a level to the next on…
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 $N$) with unit squares and $(1,m-1;t)$-combs for some fixed $m=1,2,\dots$ and $t=2,3,\dots$ that use $n$ tiles in total of which $k$ are combs. A $(1,m-1;t)$-comb is a tile composed of $t$ unit square sub-tiles (referred to as teeth) placed so that each tooth is separated from the next by a gap of width $m-1$. We show that the entries in the triangle are co…
In two previous papers we have presented partition formulae for the Fibonacci numbers motivated by the appearance of the Fibonacci numbers in the representation theory of the 3-Kronecker quiver and its universal cover, the 3-regular tree. Here we show that the basic information can be rearranged in two triangles. They are quite similar to the Pascal triangle of the binomial coefficients, but in contrast to the additivity rule for the Pascal triangle, we now deal with additivity along hooks, or, equivalently, with additive functions for valued translation quivers. As for the Pascal triangle, we…
We consider the sequence of integers whose $n$th term has base-$p$ expansion given by the $n$th row of Pascal's triangle modulo $p$ (where $p$ is a prime number). We first present and generalize well-known relations concerning this sequence. Then, with the great help of Sloane's On-Line Encyclopedia of Integer Sequences, we show that it appears naturally as a subsequence of a $2$-regular sequence. Its study provides interesting relations and surprisingly involves odious and evil numbers, Nim-sum and even Gray codes. Furthermore, we examine similar sequences emerging from prime numbers involvin…
A circular Pascal array is a periodization of the familiar Pascal's triangle. Using simple operators defined on periodic sequences, we find a direct relationship between the ranges of the circular Pascal arrays and numbers of certain lattice paths within corridors, which are related to Dyck paths. This link provides new, short proofs of some nontrivial formulas found in the lattice-path literature.
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 $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. For every odd number $n$, we build an orbit in $\mathbb{Z}/n\mathbb{Z}$, by the linear cellular automaton generating the Pascal triangle modulo $n$, which contains infinitely many balanced Steinhaus triangles. This orbit, in $\mathbb{Z}/n\mathbb{Z}$, is obtained from an integer sequence called the universal sequence. We show…
The classical Eulerian polynomials can be expanded in the basis $t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a $q$-analogue of this expansion for Carlitz's $q$-Eulerian polynomials as well as a similar formula for Chow-Gessel's $q$-Eulerian polynomials of type $B$. We shall give some applications of these two formulae, which involve two new sequences of polynomials in the variable $q$ with positive integral coefficients. An open p…
I conjecture three identities for the determinant of adjacency matrices of graphene triangles and trapezia with Bloch (and more general) boundary conditions. For triangles, the parametric determinant is equal to the characteristic polynomial of the symmetric Pascal matrix. For trapezia it is equal to the determinant of a sub-matrix. Finally, the determinant of the tight binding matrix equals its permanent. The conjectures are supported by analytic evaluations and Mathematica, for moderate sizes. They establish connections with counting problems of partitions, lozenge tilings of hexagons, dense…
We introduce the tetrahedron trinomial coefficient transform which takes a Pascal-like arithmetical triangle to a sequence. We define a Pascal-like infinite tetrahedron H, and prove that the application of the tetrahedron trinomial transform to one face T of H provides the opposite edge E to T in H. It follows from the construction that the other directions in H parallel to E can be obtained similarly. In case of Pascal's triangle the sequence generated by the trinomial transform coincides the binomial transform of the central binomial coefficients.
We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, all of them rather involved or else relying on sophisticated number theoretical arguments. We not only give a simple and elementary proof, but also show the identity generalizes to arrays other than Pascal's triangle. As an application we obtain identities relating trinomial coefficients and Catalan's triangle to Fibonacci numbers.
The paper considers the sequence of the Motzkin words, which is constructed according to formal features of natural numbers. We investigate the decomposition of well-formed parentheses into the matched pairs of parentheses (analogous to Prime Numbers). The point triangle of bracket pairs is constructed and studied, as well as the numerical Motzkin triangle (similar to Pascal's triangle). To calculate the Motzkin triangle, the graphical method of stars and bars is used.
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 linear recurrent sequences. We determine the sums and alternating sums of the elements in columns, and we give some examples of the trinomial transform triangle.
The binomial interpolated transform of a sequence is a generalization of the well-known binomial transform. We examine a Pascal-like triangle, on which a binomial interpolated transform works between the left and right diagonals, focusing on binary recurrences. We give the sums of the elements in rows and in rising diagonals, further we define two special classes of these arithmetical triangles.
The summation formula within pascalian triangle resulting in the fibonacci sequence is extended to the $q$-binomial coefficients $q$-gaussian triangles.
