组合序列的正性研究

Reasearch on the positivities is one of the important topics in Combinatorics, which has many important connections with Analytics and Algebra. The object of this project is to study various positivities of combinatorial sequences by combining combinatorial methods, analytic skills and algebraic techniques. It mainly includes the following:.1. \gamma positivity problems of combinatorial sequences. It refers to many major problems in Combinatorics. We will study the \gamma positivity problems by means of combinatorial methods and the theory of the symmetric functions..2. Log-concavity and log-convexity of combinatorial sequences. Research on log-concavity and log-convexity is a significant topic in positivity. We will use some classical methods, the theory of the symmetric functions and the skills of the multivariate stable polynomials to research some log-concavity problems. With the help of the theory of operator, study the infinite (q-) log-concavity and log-convexity of some combinatorial sequences..3. Real rootedness of polynomials. The research on polynomials with only real zeros plays an important role in positivity. The project will study some open problems on polynomials with only real zeros by combining the theory of total positivity of matrices and analytic skills of the multivariate stable polynomials...We hope the results of this project will have positive effect and play a promoting role in positivity of combinatorial sequences.

组合序列的正性是组合数学的重要研究课题，与分析、代数等数学分支有着密切的联系。本项目将组合方法与分析工具、代数技巧等相结合，研究组合序列的各种正性性质，主要研究内容包括.1. 组合序列的\gamma正性问题。其涉及到组合学中多个重要的公开问题。本项目将运用组合方法及对称函数理论等研究组合序列的\gamma正性问题。.2. 组合序列的对数凹凸性。对数凹凸性是组合序列正性研究的热点课题。本项目将采用一些经典方法、对称函数理论及多变量stable多项式手段，研究一些组合序列的对数凹性猜想，运用算子理论等代数方法研究组合序列的无穷(q-)对数凹凸性问题。.3. 多项式的实零点性。实零点多项式的研究对正性问题的解决起着重要作用。本项目将综合运用矩阵的正性理论及多变量stable多项式技巧，研究组合学中一些实零点多项式的问题。.希望本项目的研究成果将对组合序列的正性研究产生积极影响和推动作用。

组合序列的正性问题，如单峰性、对数凹凸性及其q模拟、Stieltjes Moment性等，是组合数学中的重要研究课题，它与分析、代数与几何、概率与统计等数学分支密切联系。.在该项目中，课题组对组合序列的正性问题做了系统研究，主要成果有：.1.Stieltjes Moment问题是分析中的重要研究课题。我们建立了经典分析中Stieltjes Moment问题与组合序列无穷对数凸性的关系，证明了Stieltjes Moment具有无穷对数凸性；提出了q-Stieltjes Moment的概念，并用来给出了三角卷积保持Stieltjes Moment性质的标准；证明了许多组合多项式都是q-Stieltjes Moment序列。.2.Q-模拟是组合学研究的重要课题。我们给出了强-q对数凸性的算子方法与结果，得到了3-q对数凸性的标准，从而推广和加强了q对数凸性的许多已知结果。.3.实零点多项式的研究对正性问题的解决起着重要作用。我们发展了判断独立多项式仅有实零点性等单峰型性质的算子方法，统一处理了许多已知结果；我们统一证明了许多满足四项递推关系的组合三角行发生函数仅有实零点性。.我们的研究成果丰富了组合数学中正性问题的研究。