Research On The Distribution Properties Of Several Types Of Polynomial Sequences In Combinatorics

The distribution property of combinatorial is a basic problem in combinatorics.The property arises often in combinatorics,algebra,probability,statistics and other branches of mathematics.It also arises in computer sciences,economics and other science.One of the most important distribution property is unimodal properties which includes uni-modality,log-concave,log-convex and reality of zeros property.The unimodal properties have been widely studied.People also pay attention to the normal distribution of some combinatorial sequences.Normal distribution is an important distribution in probability and statistics.There is close relation between polynomials with nonnegative coefficients and normal distribution.Therefore,the unimodal properties and normal distribution of some combinatorial sequences have research value.Algebraic graph theory is an important branch which is applied algebra to study the problem of graph theory.Graph polynomials theory is the kernel of the algebraic graph theory which contains graph polynomials sequence.Graph polynomial sequences have important combinatorial significance.In 1997,Benoumhani proposed two kinds of poly-nomials related to the second kind of Whitney numbers of Dowling lattice,then a series of important theorems and conclusions have been obtained since many scholars have paid attention to it.In this paper we study the unimodal properties of the adjoint polynomials and edge cover polynomials,including unimodality,log-concavity and reality of zeros.We also give the asymptotic normality of coefficients sequence of some polynomials related to Dowling lattices.In this article we mainly study two parts.The first part discusses the unimodal properties of adjoint polynomials and edge cover polynomials.In graph theory,chromatic polynomial is a function of chromatic number which used to calculate the number of methods of graph coloring.In 1912,it was proposed by Birkhoff to conquer the Four-Color Conjecture.The adjoint polynomial was introduced to solve the problem of the chromatic polynomials of complementary graph.The edge cover polynomials was introduced to explore the notion of the vertex cover polynomials.The two kinds of graph polynomials have a great impact on graph theory and many applications.At first,we give the expression of adjoint polynomials of these kinds of graphs by establishing the recurrence relations of the polynomials.Then we give the sufficient conditions for the operation preserving the symmetry of graph polynomials.As applications,we obtain that the unimodal properties of edge cover polynomials.The second part discusses the asymptotic normality of coefficients sequence of some polynomials.Dowling constructed a class of geometric lattices over a finite group G of order m?1,denoted by Q_n?G?.When m=1,G regard as the trivial group,Q_n?G?is isomorphic to the lattice of partitions of an?n+1?element set,which we denote by?_n+1.So Dowling lattices can be viewed as a group theoretic analog of partition lat-tices.The B_n?x,y,z?is the generalization form of Bell polynomials which contains some combinatorial polynomial related to the Dowling lattice.So we research the properties of the B_n?x,y,z?and then generalize to other polynomials.At first,we show that the asymptotic normality of B_n?x,y,z?sequences.As applications,we obtain uniformly that the asymptotic normality of coefficients sequence of polynomials related to Dowling lat-tices,such as Bell polynomials,r-Bell polynomials,Dowling polynomials and r-Dowling polynomials.