Nested Chain Decompositions For Posets

Sperner theory is a very lively area of combinatorins. The main content of Sperner theory is to investgate the extremal problems on posets. For rank-unimodal posets, Griggs conjectured that normal matching implies nested chain partitions in 1977.In this thesis, we investigate nested chain partitions for posets.The organization of this paper is as follows:In the first chapter, we review some basic terminology and methods in Sperner theory.In the second chapter, we show the importance of rank 3 posets for Griggs conjecture and mainly consider four approaches to nested chain partitions for rank 3 posets. We also verify this conjecture for special posets of two kinds.In the following two chapters, we investigate symmetric chain partitions for important posets of two kinds.In the third chapter, we give an algorithm to generate Hasse diagram of poset L(m, n), and mainly investigate symmetric chain partitions for L(3, n).In the forth chapter, we firstly give an algorithm to generate necklace poset Nn from subset lattice Bn,and then investigate the symmetric chain partitions for Nn.