Several Research Proplems In Enumerative Combinatorics

Enumerative Combinatorics is one of vitally important research branches in Combinatorics, mainly investigates counting problems of combinatorial settings on finite set under given conditions. The main contents of this discourse are listed as follows:In the first chapter, we define two families of generalized p-Stirling numbers which unifies the binomial coefficients and the classical Stirling numbers. We discuss the combinatorial interpretation of p-Stirling numbers and extend the one-dimensional finite set partitions and permutations to general p-dimensional cases;obtain several difference identities of closed form of p-Stirling numbers and investigate the determinantal properties of p-Stirling matrices .In the second chapter, we work over a simple but important combinatorial structure — Dyck paths, which is a hot research topic in recent decades. First, we describe the connections between the set of Dyck paths with strictly increasing valleys and the set of compositions of an integer;then bijections and the methods of generating trees together with those of Riordan arrays are used to enumerate these subsets of D_m, resulting in such well-known sequences as the Catalan nos., Narayana nos., Motzkin nos., Fibonacci nos., Schroder nos., and the unsigned Stirling numbers of the first kind. In particular, we gives two configurations that apparently do not appear in Stanley's well-known list of Catalan structures. Finally, we define a new kind of restricted permutation pattern and discuss the connection between the set of associated Dyck paths and the set of restricted permutations.In the third chapter, we study the algebraic properties of generalized Fibonacci polynomials, including the properties of matrices composed of their coefficients, the combinatorial meanings of their coefficients and the convolution summation formulas of generalized Fibonacci polynomials of the general type.In the last chapter, based on the technique of MacMahon's partitions analysis, we generalize Sellers' theorem on partitions to a more general case and give several applications, involving many classical sequnences such as Bell numbers, Fibonacci numbers, Lucas numbers, and Pell numbers;We also investigate the formulas of R(N) of the number of partitions of an integer N into distinct Fibonacci numbers by a transformation on the binary representation and present several recursive formulas in different forms, by which we can easily obtain R(N) for any large number TV.