基于序列比率的组合性质研究

序列的组合性质理论是当前组合数学的研究热点之一。该课题与其它多个数学分支有着重要的联系，吸引了包括国际数学家大会一小时报告人Richard P. Stanley教授在内的众多知名组合学家的研究兴趣。. 本项目旨在利用数学机械化方法和代数组合理论从序列前后项比率的角度研究序列的组合性质。利用数学机械化方法，我们将确定Boros-Moll序列前后项比率的上界和下界并利用上述上界和下界，细致刻画Boros-Moll序列的组合结构；利用代数组合手段，我们将系统研究比率单调性和交错对数凹性在某些算子作用下的不变性理论；我们将尝试把数学机械化方法和组合分析结合起来，构造中间函数,寻找新的算法，计算递归关系，研究无穷序列的组合性质。.在项目的实施过程中，我们期望利用申请书中的方法，不仅能够发现这些序列满足的组合性质，刻画出序列的组合结构，而且能进一步深化该领域与数学机械化方法的联系。

The theory of combinatorial properties of sequences is one of the most active fields in contemporary combinatorics, which has numerous natural connections to other branches of mathematics, that is exactly the reason why many well-known combinatorialists, including Professor Richard P. Stanley who gave a plenary lecture in the International Congress of Mathematics, are interested in it..This project focuses on using the theories of mathematics mechanization and algebraic combinatorics, starting from studying the ratios of sequences, to investigate combinatorial properties of sequences. Applying the methods of mathematics mechanization, we aim to determine the upper bound and lower bound of the ratios of any two successive terms in the Boros-Moll sequences, based on which, we shall characterize the combinatorial structures of the Boros-Moll sequences. Furthermore, the methods in algebraic combinatorics allow us to study the invariable theories of the ratio monotonicity and the interlacing log-concavity under the action of certain operators. Our ultimate purpose is to investigate the combinatorial properties of infinite sequences, for which we combine the mathematics mechanization and analytic combinatorics together, to derive the recurrence relations of the terms via constructing intermediate functions and finding new algorithms..In the process of executing this project, we anticipate that by the methods mentioned in this project, we can not only discover the underlying combinatorial properties of these sequences, but also describe the combinatorial structures of these sequences and eventually intensify the bonds to mathematics mechanization.