全正性理论在组合数学中发挥着越来越重要的作用。本项目研究组合数学中的一些全正性问题，包括组合矩阵的全正性、组合数列和多项式序列的Toeplitz全正性和Hankel全正性。研究内容围绕三类重要组合三角的全正性展开。.1. Riordan三角的全正性：Riordan三角有各种各样的刻画，按矩阵乘法也构成一个群。我们将从不同角度给出Riordan三角全正性的一些判断法则。.2. Aigner三角的全正性：根据Aigner三角元素的格路计数背景，给出Aigner三角全正性的组合解释。证明Sokal关于Apéry数和Apéry多项式的Hankel全正性猜想。.3. Eulerian型三角的全正性：研究在Eulerian型三角中，行数列的Toeplitz全正性和渐近正态性。证明Brenti关于Eulerian三角全正性的猜想。

The theory of total positivity plays an increasing important role in combinatorics. The objective of this project is to investigate some combinatorial problems related to total positivity, including the total positivity of combinatorial matrices, the Toeplitz- and Hankel-total positivity of counting coefficients and polynomial sequences respectively. The content of the project is divided into three parts..1. Total positivity of Riordan arrays. There have been miscellaneous characterizations of Riordan arrays and we will give some criterions for total positivity of Riordan arrays from various viewpoints..2. Total positivity of Aigner matrices. We will give combinatorial interpretations for totally positive Aigner triangles based on lattice path techniques. We also study Sokal's conjectures on Hankel-total positivity of Apéry numbers and polynomials..3. Total positivity of Eulerian-type triangles. We will investigate Toeplitz-total positivity and asymptotic normality of rows in Eulerian-type triangles. We also hope to prove Brenti's conjecture on the total positivity of the Eulerian triangle.