Asymptotic Normality Of Several Combinatorial Sequences

In 1952,Ivan Lah introduced the definition of(signless)Lah number,then many scholars generalized this Lah numbers to the associated Lah number,the ordered Lah number and the s-associated Lah number,and obtained many interesting results.Benoumhani introduced two kinds of generalized Dowling polynomials,and some important properties related to these polynomials have been obtained,which show that Dowling lattices have a very broad research prospect.Normal distribution is a common probability distribution in mathematics and plays an important role in algorithm analysis.In recent years,the asymptotic normal distribution of combinatorial sequences has excited the interest of many scholars.In this dissertation,we study the asymptotic normality of generalized Lah numbers and coefficient sequence of polynomials related to Dowling lattice by Harper's method.The method divides the process of proof into two steps:The first step is to prove that the generating function of the sequence has only real zeros;and the second step is to prove that the variance related to the sequence tends to be infinity.The details of this dissertation are as follows:In the first part,we mainly discuss the asymptotic normality of the generalized Lah numbers and some important properties of the(signless)Lah number.In this section,we show the asymptotic normality of the associated Lah number,the ordered Lah number,and the 2-associated Lah number.As application,we obtain the asymptotic normality of the(signless)Lah number.In addition,we also show the total positivity of the(signless)Lah triangle and the strict total positivity of the(signless)Lah square.In the second part,we mainly consider the asymptotic normality of coefficient sequences of several polynomials related to Dowling lattice.The polynomial F_n(x,y,z)generalizes the ordered Bell polynomial and the Benoumhani generalized Dowling polynomial.In this sec-tion,we prove the asymptotic normality of coefficient sequence of the polynomial F_n(x,y,z)firstly.As applications,we obtain the asymptotic normality of coefficient sequence of the Benoumhani generalized Dowling polynomial and the ordered Bell polynomial.