组合统计量与构造性组合方法的研究

排列、树和匹配等组合结构的统计量是代数组合学的重要研究对象，并且经常出现在多面体几何学和代数拓扑学的研究中。本项目旨在利用构造性组合方法研究有关排列、标号树和完美匹配的几个问题。具体内容包括：(1)研究双gamma非负性，着重研究对称分解模型，这方面的进展可以促进多面体细分问题的研究。双gamma非负性蕴含着交错递增性，gamma非负性是双gamma非负性的退化情形；(2)为了更深刻地揭示群作用下的不变统计量，将从统计量分解的角度出发在有限Coxeter群中的排列和标号树上研究包含succession统计量的联合分布；(3)从组合算法的角度出发研究Stirling排列与对称群中的排列以及完美匹配的内在联系，这可以加深对组合统计量的理解。本项目将充分结合组合算法、文法理论、连分式理论以及拟对称函数理论，通过新的视角审视几个前沿热点问题，做出具有特色的工作，争取在若干重要问题上取得进展。

The statistics of permutations, trees, matchings and so on are important research objects in algebraic combinatorics, which also appear frequently in polyhedral geometry and algebraic topology. The purpose of this project is to study some problems related to permutations, labeled trees and perfect matchings by using constructive combinatorial methods. The main contents are as follows: (1)To study bi-gamma positivity, especially on the symmetric decomposition models, which will promote the study of subdivision problems of polytopes. It should be noted that bi-gamma positivity implies alternatingly increasing property and gamma positivity is a reduced form of bi-gamma positivity; (2)To study the joint distributions including succession statistics over finite Coxeter groups and labeled trees from the viewpoint of the decomposition of statistics, so as to reveal much deeper invariant statistics under group actions; (3)To explore connections among Stirling permutations, permutations in the symmetric group and perfect mathchings from the viewpoint of combinatorial algorithms, so as to better understand combinatorial statistics. We will combine combinatorial algorithms and the theories of grammars, continued fractions and quasisymmetric functions, and try to make progress on some important problems.