代数组合学是组合数学的重要研究分支。近些年，表示论、交换代数、代数几何中的工具和方法被不断引入用于研究代数组合中的问题。本项目主要研究Schubert计数演算（Schubert’s Enumerative Calculus）的组合性质，这是代数组合和代数几何紧密交叉、国际上备受关注的一个研究课题，代表了代数组合的国际前沿方向。从组合学的角度，Schubert计数演算涉及的核心结构是Schubert多项式。Schubert多项式包含Schur函数为特殊情形，与对称函数理论有密切联系。本项目拟研究问题主要包括：建立Schubert多项式不同组合模型之间的联系；组合证明Schubert多项式的Macdonald公式；研究Schubert多项式的Merzon-Smirnov猜想；发掘特殊类型的Schubert多项式的组合公式。

Algebraic combinatorics is an important research area of combinatorics. In recent years, tools and methods from representation theory, commutative algebra and algebraic geometry are introduced to study problems in algebraic combinatorics. In this project, we shall aim to study the combinatorial aspects of Schubert's Enumerative Calculus, which have been an project in the intersection of algebraic combinatorics and algebraic geometry, and represent the frontier of algebraic combinatorics. From the combinatorial point of view, the key structures of Schubert's Enumerative Calculus are Schubert polynomials. Schubert polynomials include Schur functions as a special case, and are closely related to the theory of symmetric functions. Problems that we are concerned with in this project mainly include: establish relations between different combinatorial models of Schubert polynomials; present a combinatorial proof of the Macdonald's formula on Schubert polynomials; investigate the Merzon-Smirnov conjecture on Schubert polynomials; explore combinatorial formulas for specific classes of Schubert polynomials.