相关论文: Counting on rectangular areas

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…

As a first application of a very old theorem, known as Herschel's theorem, we provide direct elementary proofs of several explicit expressions for some numbers and polynomials that are known in combinatorics. The second application deals…

In this paper we introduce the concept of the rectangular metric like spaces, along with its topology and we prove some fixed point theorems under different contraction principles. We introduce the concept of modified metric-like space as…

A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…

We present algorithms for classification of linear codes over finite fields, based on canonical augmentation and on lattice point enumeration. We apply these algorithms to obtain classification results over fields with 2, 3 and 4 elements.…

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…

We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces,…

For an arbitrary prime $p$ we prove that the proportion of entries divisible by $p$ in certain columns of the character table of the symmetric group $S_n$ tends to 1 as $n\to\infty$. This is done by finding lower bounds on the number of…

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…

Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of…

This paper concerns the self-similarity of topological spaces, in the sense defined in math.DS/0411344. I show how to recognize self-similar spaces, or more precisely, universal solutions of self-similarity systems. Examples include the…

We introduce and study properties of certain new multifunctional harmonic spaces in the upper halfspace.We prove several sharp embedding theorems for such multifunctional spaces,these results are new even in the case of a single function.

It is shown that rational points over finite fields of moduli spaces of stable quiver representations are counted by polynomials with integer coefficients. These polynomials are constructed recursively using an identity in the Hall algebra…

A Central Limit Theorem is proved for linear random fields when sums are taken over finite disjoint union of rectangles. The approach does not rely upon the use of Beveridge Nelson decomposition and the conditions needed are similar to…

We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common…

We give an algorithmic proof of Pick's theorem which calculates the area of a lattice-polygon in terms of the lattice-points.

Combinatorial categories satisfy a stronger form of Yoneda Lemma, namely, the isomorphism type of an object can be recovered by counting the number of homomorphisms from all other objects into it. In this work, we show that this property…

We describe the canonical correspondence between set of all finite metric spaces and set of special symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those…

In the context of generating uniform random contingency tables with pre-specified marginals, the number of (binary) matrices with given row- and column-sums is a well-studied object in the literature. We will denote this number by $N(p,q)$,…