Combinatorics On Lattice Paths

Lattice paths are one of the classical objects in combinatorics, which have re-ceived a great deal of attention and are closely related to other combinatorial objects. In this paper, we are mainly concerned with the Hankel determinants and Chung-Feller properties on lattice paths.Let{al}l≥O be a sequence. For a nonnegative integer k, let Ank denote the Hankel matrix of order n of the sequences{al}l>O of the form Ank={a-k+i+j-2)ni,j=1, where {al}l>o could be Catalan, Motzkin and Schroder numbers. The problem of evaluating the determinant Ank has been extensively studied. For example, it is well known that det1≤i,j≥n(ci+j-2)=1, det1≤i,j≤n(ci+j-1)=1 and det1≤i,j≤n(Ci+j)=n+1; Cameron and Yip showed that the Hankel determinant of sums of consecutive Motzkin numbers are closely related to a Chebychev polynomial of the second kind; Rajkovic, Petkovic and Barry obtained the explicit formula for the Hankel determinant of sums of con-secutive Schroder numbers using orthogonal polynomials; Eu, Wong and Yen derived the generating functions and explicit formulae for the Hankel determinants of the se-quence of linear combinations of two consecutive weighted Schroder paths. The result that they derived is based on the well-known Gessel-Viennot-Lindstrom lemma,which is the basic theory in this paper.The famous Chung-Feller theorem was first discovered by MacMahon in 1909; In 1949, Chung and Feller proved this theorem by using analytic method and named it by their names, then, it was proved again by Narayana et al.with cyclic paths; In 2005, Eu, Fu and Yeh refined this theorem by studying the Taylor expansions of generation functions of different lattice paths; And in 2007, by introducing the notion of butterfly decomposition of double root trees, Chen et al. provide an alternative proof of the Chung-Feller properties of lattice paths obtained by Eu, Fu and Yeh.In this thesis, we are mainly concerned with the Hankel determinants and Chung-Feller properties on lattice paths. This thesis consists of three chapters.In Chapter 1, we introduce some basic definitions and notations, give a chief sur-vey in this direction and state the main results obtained.In Chapter 2, we derive the generating functions and explicit formulae for the Hankel determinants of weighted Motzkin numbers, weighted free Motzkin numbers and weighted Schroder numbers.In Chapter 3, we obtain Chung-Feller properties of a class of lattice paths in an-swer to a problem posed by Dziemianczuk.