Combinatorial Proofs Of Log-convexity And Log-concavity Of Some Combinatorial Numbers

It is interesting to know the distirbution of combinaotirial sequences: log-convexity, log-concavity. This is a fertile source of inequalities, which are paricularly useful in estimates. Log-convex sequences and log-concave sequences arise often in combiatorics, algebra, analysis, geometry, computer science, probability and statistic. There has been an amount of interest and research devoted to this topic in recent year. In 1989, Stanley gave an excellent survey of the log-concavity. Very recently, Liu and Wang gave a survy of the log-convexity. In general, the tools that are useful in proving the log-convexity and log-concavity include classic analysis, linear algebra, the representation theory of Lie algebras, algebraic geometry and the theory of total positivity. Since our main interest stems from the combinatorial motivations, we always look for the combinatorial interpretation for the log-convexity and the log-concavity. In this thesis, we show how lattice paths, Dyck paths and the Reflection Principle can be used to give combinatorial proofs of log-convexity and log-concavity of sequences, as well as q-log-convexity and q-log-concavity. We obtain some new linear transformation preserving the log-convexity.In the first chapter, we review severial basic definitions and results on the log-convexity and log-concavity.In the second chapter, we give some combinatorial proof of log-convexity and log-concavity of some combinatorial numbers by lattice paths, Dyck paths and the Reflection Principle. In Section 2.1, we provide the combinatorial proofs of some combinatorial numbers, such as the large Schroder numbers, the central Delannoy numbers and the Fine numbers. In Section 2.2, we prove the log-concavity of the Delannoy numbers and present the generalization of the conjecture of Simion.In the third chapter, we give some q-log-convex and q-log-concave results and some new linear transformations preserving log-convexity...