Enumerative Properties And Application Of The Several Classes Of Matchable Distributive Lattices

In order to studying the integer property of all matchings,Z-transformation graph(also called resonance graph)is established on the perfect matching set of plane bipartite graphs,then the distributive lattice structure on the perfect matching set is given by orientation,it is also proved that the directed or undirected Hasse diagram of the distributive lattice is isomorphism the directed or undirected Z-transformation graph of the perfect matching set.In this paper,we study some enumerative properties of several classes matching distributive lattices of some posets similar to "Zigzag poset".There are four chapters in this paper.In the first chapter,we introduce the main background of the article,some preliminaries,special marks and some main conclusions.In Chapter 2,on the basis of the Fibonacci cubes which was studied by Hsu,and the structure of Fibonacci cube be considered,besides,there are some enumerative properties about Fibonacci cube are introduced.we consider the outdegree polynomial and indegree polynomial of Fibonacci cube.In Chapter 3,we give a new poset which is similar to "Zigzag poset ",called "S-fence",and the under-lying graph of the directed Hasse diagram of the filter lattice of S-fence named Fibonacci-like cube according to the number of the vertices,the enumerative properties and several classes of polynomial are obtained of Fibonacci-like cube such as rank generating function,cube polynomials,maximal cube polynomials,degree sequence polynomials and indegree sequence polynomials.Besides,we found some special results about the binomial coefficients and the Padovan sequence.In Chapter 4,we structured a special hexagonal chain which is similar to the "Zigzag hexagonal chain" named "silucasence",and give a special poset the Hasse diagram is isomorphic to the inner dual graph of silucasence,and called "Y-fence",the underlying graph of Hasse diagram of its filter lattice and the filter lattice are called Lucas-like cube,we get some fundamental properties and several polynomials about the Lucas-like cube,e.g.rank generation function,cube polynomials,maximal cube polynomials and degree sequence polynomials.