2019.10.17
2:00-3:00 Lecture 1
Title : Combinatorial Enumeration
Abstract: In this talk, we present a survey of results on Chen’s grammars (also known as context-free grammars) and its applications. By introducing the change of context-free grammars method, we first present a unified grammatical proof of the gamma-positivity of Eulerian polynomials, type B Eulerian polynomials and Narayana polynomials. We then provide partial gamma-positive expansions for several multivariate polynomials associated to Stirling permutations, Legendre-Stirling permutations, Jacobi-Stirling permutations and type B derangements, and the recurrence relations for the partial gamma-coefficients of these expansions are also obtained. Moreover, we show that the coefficient polynomials of Jacobi elliptic function sn(u,k) are all symmetric and unimodal. Furthermore, we present a survey of our results on Stirling permutations.
2019.10.18
10:00-11:00 Lecture 2
Title : Some recent progresses in studies of the Tutte polynomial of a graph
Abstract: William Tutte is one of the founders of the modern graph. For every undirected graph, Tutte defined a polynomial TG(x; y) in two variables which plays an important role in graph theory. In this talk, we will introduce some recent progresses in studies of the Tutte polynomial of a graph.
In Tutte’s original definitions, non-negative integers, called internal and external activities with respect to the arbitrary enumeration, are defined for each spanning tree, they serve as the indices of x and y in the product that is the corresponding term of Tutte polynomial. First, We will introduce the conceptions of -cut tail and -cycle tail of T, which are generalizations of the conceptions of internally and externally activities, repectively, where is a sequence on the edge set of G and T is a spanning tree of G. We will also discuss the conceptions of proper Tutte mapping and deletion-contraction mapping. In 2004, Postnikov and Shapiro introduced the concept of G-parking functions in the study of certain quotients of the polynomial ring. The Tutte polynomial of the graph G can be expressed in terms of statistics of G-parking functions. Let Δ be a nonsingular M-matrix. We will introduce Δ-parking
functions which is a generalization of G-parking functions.
We will introduce the abelian sandpile model and Δ-recurrent configurations. There is a simple bijection between Δ-parking functions and Δ-recurrent configurations. We will discuss the geometry of sandpile model. In general, the Tutte polynomial encodes information about subgraphs of G. For example, for a connected graph G, TG(1; 1) is the number of spanning trees of G, TG(2; 1) is the number of spanning forests of G, TG(1; 2) is the number of connected spanning subgraphs of G, TG(2; 2) is the number of spanning subgraphs of G. At last, we will discuss combinatorial interpretations of TG(1 + p;-1) and TG(-1; 1).
11:00-12:00 Lecture 3
Title : Lattice paths and Uniform partitions
Abstract: The classical Chung-Feller theorem gives a uniform partition of all n-Dyck paths in the plane. In this talk, we will introduce generalizations for the classical Chung-Feller theorem, discuss the functions of uniform-partition type and their combinatorial interpretations. Many uniform partitions of combinatorial structures
are consequences of the cycle lemma. In this talk, we will introduce the generalizations of the cycle lemma.