The Study Of Combinatorial Sequences By Using Grammars And Probabilistic Methods

Modern mathematics can be divided into two major categories according to the continuity and discretization of the research objects,while combinatorics is an important branch for studying discrete objects.Combinatorics has closely relationship with group theory,graph theory,computer science and other disciplines.In recent years,with the rapid development of computer science,combinatorics attracted many people.The main contents of combinatorics include combinatorial enumeration,combinatorial optimization,combinatorial identities and combinatorial matrices and so on.In this paper,we mainly uses grammars and combinatorial probability methods to study the combinatorial counting problems.In this paper,we first introduce the definition and properties of grammars.Secondly,we study signless Stirling numbers of the first kind by using grammars,and as applications,we study some associated permutation statistics.Thirdly,using the grammars,we study rooted label trees and alternating descents of permutations.Fourthly,by using combinatorial probability method,we study expectations and variances of unsign Stirling numbers of the first kind and the big Eulerian numbers.In particular,we show that the generating polynomial of big Eulerian numbers have only real zeros.Furthermore,we prove that the sequence of big Eulerian numbers satisfy the central and local limit theorems.Finally,we summary the main results and pose several questions.