Related papers: The Hockey Stick Theorems in Pascal and Trinomial …

In this paper, we will show dichotomy theorems for the computation of polynomials corresponding to evaluation of graph homomorphisms in Valiant's model. We are given a fixed graph $H$ and want to find all graphs, from some graph class,…

The Fibonacci sequence is obtained as weighted sum along the rows in the Pascal triangle by choosing a periodic up-and-down pattern of weights from the set $\{-1,-\frac{1}{2},0, \frac{1}{2}, 1\}$. A graphical illustration of this identity…

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 this paper, we state and prove some congruence properties for the trinomial coeficients, one of which is similar to the Wolstenholme's theorem.

The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.

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 Sierpi\'nski gasket that…

We show that the theorem of the three perpendiculars holds in any n-dimensional space form.

We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.

We tour several Euclidean properties of Poncelet triangles inscribed in an ellipse and circumscribing the incircle, including loci of triangle centers and envelopes of key objects. We also show that a number of degenerate behaviors are…

A variable line through the centroid G of a triangle divides the triangle into two parts each of whose lengths as a fraction of the perimeter fills a closed interval [m,1-m], with m between 0 and 1/2. We show that the range of m taken over…

We introduce and study a new notion of patterns in Stirling and $k$-Stirling permutations, which we call block patterns. We prove a general result which allows us to compute generating functions for the occurrences of various block patterns…

We present various constructions of sequences of polynomials satisfying the Binomial Theorem in finite characteristic based on the theory of additive polynomials. Various actions on these constructions are also presented. It is an open…

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…

We study line bundles on toric DM stacks $\mathbb{P}_{\mathbf{\Sigma}}$ of dimension two. We give a combinatorial criterion of when infinitely many line bundles on $\mathbb{P}_{\mathbf{\Sigma}}$ have trivial cohomology. We further discuss…

Here I present several theorems about trapezoids tilings. The first one is related to trapezoids with rational base relation, the other ones are related to those with base relation from quadratic number field.

In these notes, we present versions of trace theorems for Sobolev spaces over an interval in the real line, and also a one-dimensional version of the well-known Poincare inequality.

In this expository paper we collect some simple facts about analogues of Pascals triangle where the entries count subsets of the integers with an even or odd sum and show that they are related to Rogers-Szego polynomials. In particular we…

We introduce patterns on a triangular grid generated by paperfolding operations. We show that in case these patterns are defined using a periodic sequence of foldings, they can also be generated using substitution rules and compute…

We examine a class of geometric theorems on cyclic 2n-gons. We prove that if we take n disjoint pairs of sides, each pair separated by an even number of polygon sides, then there is a linear combination of the angles between those sides…

In this paper the problem of finding a normal form of triangles and plane quadrilaterals up to similarity is considered. Several normal forms for triangles and a normal form for quadrilaterals of special case are described. Normal forms of…