Algebraic combinatorics with applications in geometry

组合学是研究有限对象集合的排列以及它们之间的关系的学科。组合学的研究主要受到其他数学和物理科学以及工程学应用的推动。许多这样的应用只需要确定集合中对象的数量，称为对集合进行计数。在代数组合学中，我们广泛使用了代数中的结果和方法。最基本的组合对象之一称为排列，它对有限集合中的元素进行重新排序。换位是一个非常简单的排列，它简单地交换(“换位”)两个元素的顺序。排列的乘积是通过逐次重新排序得到的，在代数中，具有该乘积的集合的所有排列的集合称为对称群。

Combinatorics is the study of arrangements of sets of finite objects and relationships between them. Research in combinatorics is centrally motivated by applications in other mathematical and physical sciences, and in engineering. Many such applications require only that one determine the number of objects in a set, called "counting" the set. In algebraic combinatorics we make extensive use of results and methods from algebra in this study. One of the most fundamental combinatorial objects is called a permutation, which reorders the elements of a finite set. A transposition is a very simple permutation, that simply interchanges the order of ("transposes") two of the elements. A product of permutations is obtained by successive reordering, and in algebra, the set of all permutations of a set with this product is called the symmetric group.