Related papers: Counting monomials
In recent work, Zeilberger and the author used a functional equations approach for enumerating permutations with r occurrences of the pattern 12...k. In particular, the approach yielded a polynomial-time enumeration algorithm for any fixed…
The present review presents the authors previous results on the topic from the title in a new light. Most of the previous results were obtained using the techniques of antilinear Hilbert-Schmidt mappings of one Hilbert pace into another,…
We present a systematic technique to find explicit solutions of birational maps, provided that these solutions are given in terms of elliptic functions. The two main ingredients are: (i) application of classical addition theorems for…
The numerical Hilbert series combinatorics and the comodule Hilbert series combinatorics are introduced, and some applications are presented, including the MacMahon Master Theorem.
We present and analyze two algorithms for computing the Hilbert class polynomial $H_D$ . The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm…
We discuss the formal aspects of the factorial polynomials and of the associated series. We develop the theory using the formalism of quasi-monomials and prove the usefulness of the method for the solutions of nontrivial difference…
We study a 2-parameter family of enumerative problems over the reals. Over the complex field, these problems can be solved by Schubert calculus. In the real case the number of solutions can be different on the distinct connected components…
We propose a splitting algorithm for solving a system of composite monotone inclusions formulated in the form of the extended set of solutions in real Hilbert spaces. The resluting algorithm is a an extension of the algorithm in [4]. The…
We present integral representations of solutions to division problems involving matrices of polynomials in several complex variables. We also find estimates of the polynomial degree of the solutions by means of careful degree estimates of…
The operational calculus associated with special polynomials has proven to be a powerful tool for analyzing and simplifying their properties. This article examines the bivariate degenerate Hermite polynomials with a focus on their…
We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by…
We use the hook lengths of a partition to define two rectangular tableaux. We prove these tableaux have equal multisets of entries, first by elementary combinatorial arguments, and then using Stanley's Hook Content Formula and symmetric…
One standard approach to compute the Hilbert function of any graded module over a field is to come up with a free-resolution for the graded module and another is via a Hilbert power series which serves as a generating function. The proposed…
By combining the telescoping method with an algebraic relation, four classes of binomial moments are examined. Several explicit summation formulae are established.
We classify the Hilbert functions of bigraded algebras in $k[x_1, x_2, y_1, y_2]$ introducing a numerical function called Ferrers function.
In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in…
We show that sampling or interpolation formulas in reproducing kernel Hilbert spaces can be obtained by reproducing kernels whose dual systems form molecules, ensuring that the size profile of a function is fully reflected by the size…
Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various…
For stationary two-valued harmonic functions with H\"older regularity, we establish their Lipschitz regularity and prove that the nodal set consists of analytic hypersurfaces away from a singular set. The main tools are the Almgren…
We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic.