组合数学中的组合不等式研究

Combinatorial inequalities play an increasingly important role in combinatorics. The object of this project is to study combinatorial inequalities by combinatorial skills, analytic techniques, probabilistic methods, and algebraic approaches. Mainly includes: 1. Combinatorial inequalities in counting. With the traditional methods of combinatorial identities, recursive relations, generating functions and lattice paths, we will look for combinatorial proof of combinatorial inequalities. We will systemically deal with combinatorial inequalities in a unified viewpoint of the theory of posets. And in particular, we will give q-analog of certain important combinatorial inequalities. 2. Positivity problems in combinatorics. Many of the major open problems of algebraic combinatorics are related to positivity questions. We will characterize symmetric polynomials with nonnegative gamma-vectors by means of the theory of symmetric functions and the theory of posets. We will also study polynomials with only real zeros from the viewpoint of the theory of total positivity. 3. Monotonicity of combinatorial and number-theoretic sequences. Monotonicity is the the important mathematical Information of inequalities. The project will combine powerful analytical techniques and probabilistic methods to study monotonicity of combinatorial and number-theoretic sequences, with the emphasis on conjectures proposed by Zhi-Wei Sun et al.

组合不等式在当前组合数学的研究中发挥着越来越重要的作用。本项目将综合运用组合计数、解析技巧、概率方法和代数途径来研究组合不等式。主要包括： 1. 组合计数中的组合不等式。借助组合恒等式、递归关系、生成函数、格路技巧等组合数学中的传统方法研究组合不等式的发现、证明和组合解释。以偏序集为平台，对一些经典组合不等式进行整合。对一些重要的组合不等式进行推广（如q-模拟）。 2. 组合数学中的正性问题。代数组合学中的许多主要公开问题都与正性问题有关。本项目将从对称多项式空间和偏序集理论两个角度研究具有非负gamma向量的组合多项式的刻画。用全正性理论研究组合多项式的实零点问题。 3. 组合数列与数论数列的单调性。单调性是反映不等式的重要数学信息。本项目将结合强有力的解析技巧和概率方法研究组合数列与数论数列相应的几类数列的单调性。重点研究孙智伟等人最近提出的系列公开问题和猜想。

组合不等式是组合数学中重要的研究内容。本项目执行期间取得如下研究成果： .1. 建立了Riordan三角和Aigner递归矩阵这样两类重要组合矩阵具有全正性的充分条件，据此可以统一给出许多著名组合三角的全正性。.2. 借助Catalan-like数的格路背景，给出了许多组合数的对数凸性和moment性质的组合解释。.3. 证明了序列的Stieltjes moment性质蕴含无穷对数凸性，据此证实了陈永川等人关于大Schroder数有无穷对数凸性的猜想。.4. 建立了序列某些单调性问题与对数凹凸性的联系，证实了孙智伟提出的一系列涉及组合序列单调性的猜想。.5. 提供了图的Laplace系数具有渐近正态性的充分条件；证明了Shapiro关于Narayana数是渐近正态的猜想。

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