Related papers: Combinatorial theorems relative to a random set

A survey of recent progress in three areas of algebraic combinatorics: (1) the Saturation Conjecture for Littlewood-Richardson coefficients, (2) the n! and (n+1)^{n-1} conjectures, and (3) longest increasing subsequences of permutations.

In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.

In this article we survey some of the recent developments in the structure theory of set addition.

Using the correspondence between a cycle up-down permutation and a pair of matchings, we give a combinatorial proof of the enumeration of alternating permutations according to the given peak set.

This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.

We present an elementary combinatorial proof of the celebrated Friendship theorem. The proof involves looking at independent sets and constructing a bound on their size which forces a contradiction.

We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.

In this article the idea of random variables over the set theoretic universe is investigated. We explore what it can mean for a random set to have a specific probability of belonging to an antecedently given class of sets.

In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show…

This is a survey of recent progress in several areas of combinatorial algebra. We consider combinatorial problems about free groups, polynomial algebras, free associative and Lie algebras. Our main idea is to study automorphisms and, more…

Analogical proportions compare pairs of items (a, b) and (c, d) in terms of their differences and similarities. They play a key role in the formalization of analogical inference. The paper first discusses how to improve analogical inference…

We derive new bounds of the remainder in a combinatorial central limit theorem without assumptions on independence and existence of moments of summands. For independent random variables our theorems imply Esseen and Berry-Esseen type…

Several results about the union-closed sets conjecture are presented.

A survey of three recent developments in algebraic combinatorics: (1) the Laurent phenomenon, (2) Gromov-Witten invariants and toric Schur functions, and (3) toric h-vectors and intersection cohomology. This paper is a continuation of…

We study a natural analogue of Collatz's Conjecture for polynomials over $\mathbb{F}_2$.

Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for…

In this paper we extend previous studies of selection principles for families of open covers of sets of real numbers to also include families of countable Borel covers. The main results of the paper could be summarized as follows: 1. Some…

We study the homology of pointed sets over a partially commutative monoid.

Combinatorics is a powerful tool for dealing with relations among objectives mushroomed in the past century. However, an more important work for mathematician is to apply combinatorics to other mathematics and other sciences not merely to…

The paper reviews various arithmetic analogues of Hamiltonian systems and presents some new facts suggesting ways to relate/unify these examples.