Related papers: Counting monomials
We develop a new method of umbral nature to treat blocks of Hermite and of Hermite like polynomials as independent algebraic quantities. The Calculus we propose allows the formulation of a number of practical rules allowing significant…
These notes present an approach to obtaining the basic operations of addition and multiplication on the natural numbers in terms of elementary results about commutative monoids.
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other…
This paper presents an algorithm for computing the contraction of two-dimensional tensor networks on a square lattice; and we combine it with solving congruence equations to compute the exact enumeration (including weighted enumeration) of…
We discuss the use of methods coming from integrable systems to study problems of enumerative and algebraic combinatorics, and develop two examples: the enumeration of Alternating Sign Matrices and related combinatorial objects, and the…
The theory of harmonic based function is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.
We define a counting function that is related to the binomial coefficients. An explicit formula for this function is proved. In some particular cases, simpler explicit formuls are derived. We also derive a formula for the number of…
We give a new algorithm for merging sorted lists of monomials. Together with a projection technique we obtain a new complexity bound for the BM-algorithm.
Two approaches for determining Hilbert functions of fat point subschemes of $\mathbb P^2$ are demonstrated. A complete determination of the Hilbert functions which occur for 9 double points is given using the first approach, extending…
We introduce regular sequences and associated Koszul resolutions for monoids in the category of functors over an essentially small linear symmetric monoidal category. Next we define polynomials over such monoids. We compute the Hochschild…
Using polynomial equations to model combinatorial problems has been a popular tool both in computational combinatorics as well as an approach to proving new theorems. In this paper, we look at several combinatorics problems modeled by…
We provide some applications of a polynomial criterion for difference sets. These include counting the difference sets with specified parameters in terms of Hilbert functions, in particular a count of bent functions. We also consider the…
The MultiplicitySequence package for Macaulay2 computes the multiplicity sequence of a graded ideal in a standard graded ring over a field, as well as several invariants of monomial ideals related to integral dependence. We discuss two…
We extend the shell and kernel reductions for hyperexponential functions over the field of rational functions to a monomial extension. Both of the reductions are incorporated into one algorithm. As an application, we present an additive…
Many algorithms have been developed for enumerating various combinatorial objects in time exponentially less than the number of objects. Two common classes of algorithms are dynamic programming and the transfer matrix method. This paper…
We present an algorithm based on continuation techniques that can be applied to solve numerically minimization problems with equality constraints. We focus on problems with a great number of local minima which are hard to obtain by local…
In the first section of this paper we prove a theorem for the number of columns of a rectangular area that are identical to the given one. In the next section we apply this theorem to derive several combinatorial identities by counting…
In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform numerical experiments suggesting a very promising efficiency. On the…
The NumericalHilbert package for Macaulay2 includes algorithms for computing local dual spaces of polynomial ideals, and related local combinatorial data about its scheme structure. These techniques are numerically stable, and can be used…
A survey written for the upcoming "Handbook of Enumerative Combinatorics".