Related papers: Tuenter polynomials and a Catalan triangle
This is a report on a talk to high school mathematics students. It gives some combinatorial connections for the entries of the boustrophedon triangle whose sides are the Taylor coefficients for the tangent and secant functions.
We describe arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects specified as a Haskell type class. Our algorithms work on a {\em generic} representation that we illustrate on…
We give a method to realize Verdier quotients as triangulated subfactors of an arbitrary triangulated category. We show that Iyama-Yoshino triangulated subfactors are Verdier quotients under suitable conditions.
In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials and more generally of the higher-order degenerate Bernoulli polynomials. We derive explicit formulas with the help of…
The Peters polynomials are a generalization of Boole polynomials. In this paper, we consider Peters and poly-Cauchy mixed type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally,…
The aim of this paper is to represent any polynomial in terms of the degenerate Frobenius-Euler polynomials and more generally of the higher-order degenerate Frobenius-Euler polynomials. We derive explicit formulas with the help of umbral…
In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…
The Eulerian polynomials enumerate permutations according to their number of descents. We initiate the study of descent polynomials over Cayley permutations, which we call Caylerian polynomials. Some classical results are generalized by…
We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of…
This paper investigates defining equations for secant varieties of the variety of reducible polynomials, which geometrically encode the notions of strength and slice rank of homogeneous polynomials. We present three main results. First, we…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
We study polynomial SL-invariants of tensors, mainly focusing on fundamental invariants which are of smallest degrees. In particular, we prove that certain 3-dimensional analogue of the Alon--Tarsi conjecture on Latin cubes considered…
It is well known that the Catalan number C_n counts dissections of a regular (n+2)-gon into triangles. Here we count such dissections by number of triangles that contain two sides of the polygon among their three edges, leading to a…
In this note, we develop transformation formulae and expansions for the log tangent integral, which are then used to derive series acceleration formulae for certain values of Dirichlet L-functions, such as Catalan's constant. The formulae…
The Macdonald polynomials expanded in terms of a modified Schur function basis have coefficients called the $q,t$-Kostka polynomials. We define operators to build standard tableaux and show that they are equivalent to creation operators…
We study the Coxeter polynomials associated with certain star-like trees. In particular, we exhibit large Salem factors of these polynomials and give convergence properties of their dominant roots.
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
We study the structure of the Fourier coefficients of low degree multivariate polynomials over finite fields. We consider three properties: (i) the number of nonzero Fourier coefficients; (ii) the sum of the absolute value of the Fourier…
Coxeter polynomials are important homological invariants that are defined for a large class of finite-dimensional algebras. It is of particular interest to develop methods to compute these polynomials. We define the notion of insertion of a…
We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In…