The theory of combinatorial models is a very important branch in combinatory and it is very powerful to solve various problems easily and elegantly and in a unified systematic way that can be used for various types.Many results in the theory of generating functions naturally come form the theory of combinatorial models and the different techniques of generating functions from different subjects in turn embellish the theory of combinatorial models. They are: finite group theory, representation theory, commutative algebra and algebraic geometry, matrices, algebraic topology, species , probabilities and statistical physics and graph theory. All those subject make combinatorics play an important role in science. In the near future, We would like to emphasize my research at relations among combinatorics models and their applications.
In this semester, we will forcus on the following three directions.
1. Combinatorial paths models. Gessel and Viennot’s paper about combinatorial paths paper have a very close relation with matrix theory and we would like to study many different combinatorial paths problems, the generating functions for many parameters of combinatorial structures and give explanations from a combinatorial point of view. For example, study the valleys and picks of combinatorial paths and fix the number of steps that are flaws against a given rule, etc. We also would like to study the mean, variance and limit distribution of random variables which correspond to combinatorial sequences.
2. Parking functions. A parking lot has exactly n parking places in a line, labeled 1, n. Those cars will enter the parking lot to park. Each car driver has a preferred number for his parking place. Let a1, …, an be those numbers, 1 <= a_i <= n. We say that the sequence a_1,…,a_n is a parking function. Each car goes to its preferred place, if it is occupied, i twill go to the next one, if the next one is occupied, then it will go to the next one after the next, etc. If no place is available, it should go out and is not allowed to come back. If all n cars can park then we call the sequence a_1, …, a_n a perfect parking function. If exactly k cars cannot park, then we call a_1,…,a_n a k-flawed parking function. If a_1= < a_2 =<…<=a_n we call it an ordered parking function. We will study study enumerative problems for the k-flawed parking functions and k-flawed ordered parking function and other different types of parking functions. We will study bijections between parking functions and other types of combinatorial structure models and with these bijections find recurrence relations and asymptotical behavior for sequences which count the number of parking functions with a given property. We also would like to study the mean, variance and limit distribution of random variables which correspond to sequences arising from parking functions problems.
3. The theory of P-partitions and its applications. Richard Stanley invented the theory of P-partitions in his Ph.D. thesis in 1971. This theory has very nice applications on graph theory, posets,.. We will study the application s of the theory of P-partitions applied on posets : Cohen-Macaulay posets, associated simplicial complexes, f-vectors, h-vectors, order polynomials, zeta-polynomials, etc. We would also like to find a reciprocity for pairs of generating functions which come from P-partitions and strict P-partitions.