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Motivated by the effective impact of the Pascal functional and the Wronskian matrices, we investigate several identities and differential equation for the Sheffer-Appell polynomial sequence by using matrix algebra. The matrix approach,…

Let $c_n$ denote the number of nodes at a distance $n$ from the root of a rooted tree. A criterion for proving the rationality and computing the rational generating function of the sequence $\{c_n\}$ is described. This criterion is applied…

We use production matrices to count several classes of geometric graphs. We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another way of counting the number of…

In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The…

We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids.

Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well…

We investigate the sequence $(P_{n}(z))_{n=0}^{\infty}$ of random polynomials generated by the three-term recurrence relation $P_{n+1}(z)=z P_{n}(z)-a_{n} P_{n-1}(z)$, $n\geq 1$, with initial conditions $P_{\ell}(z)=z^{\ell}$, $\ell=0, 1$,…

We consider a class of lattice paths with certain restrictions on their ascents and down steps and use them as building blocks to construct various families of Dyck paths. We let every building block $P_j$ take on $c_j$ colors and count all…

We study a generating function for the sum over fatgraphs with specified valences of vertices and faces, inversely weighted by the order of their symmetry group. A compact expression is found for general (i.e. non necessarily connected)…

We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these…

Let G be a graph. The black-white polynomial W_G(t) enumerates colorings of the vertices of G with two colors (black and white), where the power of t keeps track of how many white vertices have an even number of black neighbors. Such…

We describe an algorithm, implemented in Python, which can enumerate any permutation class with polynomial enumeration from a structural description of the class. In particular, this allows us to find formulas for the number of permutations…

For a sequence of polynomials $\{p_k(t)\}$ in one real or complex variable, where $p_k$ has degree $k$, for $k\ge 0$, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients $d(n,m,k)$,…

In this note we study some of the properties of the generating polynomial for homomorphisms from a graph to at complete weighted graph on $q$ vertices. We discuss how this polynomial relates to a long list of other well known graph…

In the present paper we generate binary pseudorandom sequences using generalized polynomials. A generalized polynomial is a function in whose description we not only allow addition and product (as it is the case in usual polynomials) but…

We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by…

We study the polynomial algebra (over a ring containing the rationals) in an n by m matrix of variables, and subject to the relation that says that the product of any two variables in the same column is zero. We show that the sub-algebra of…

A family of polynomials parameterized by the conjugacy classes of a finite Coxeter group is investigated. These polynomials, together with the character table of the group, determine the associated generic degrees. The polynomials are…

Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left…

This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…