排列模式的研究

排列模式是组合学理论的主要研究对象之一，且在Schubert变量的奇异性、Chebyshev多项式和很多排序算法中有重要应用。不仅如此，近年来，其与计算机科学、统计力学、计算机生物学等数学之外的其他学科之间也联系越来越密切。因此与排列模式相关的研究是当前组合数学中一个非常活跃的方向。本项目将结合组合双射、词上的组合学、算法、生成函数等来研究与排列模式紧密相关的两类问题：（1）moon polyomino填充的最长递增和递减序列；（2）词上的Wilf-型等价关系。本项目的研究内容，有的是申请人与合作者前期研究结果（发表在JCTA、AAM等）的深化和拓展，有的是国际学术界知名专家提出的公开问题。这些问题的解决，将不仅极大地丰富和完善计数与代数组合学中的相关内容，同时也将有力地推动组合学和相关学科的发展。

Permutation pattern is one of the fundamental research topics in combinatorial theory, which also appears to be frequently applied in singularities of Schubert varieties, Chebyshev polynomials and various sorting algorithms. Moreover, in recent years, due to the closer and closer connection between permutation patterns and other disciplines such as computer science, statistical mechanics, computer biology and so on, the research relevant to it becomes a very active field. This project will use combinatorial bijections, combinatorics on words, algorithms and generating function, etc. to study two closely connected topics in permutation patterns: (1) The maximal increasing and decreasing sequences in fillings of moon polyominoes; (2) The Wilf-type equivalences relations among words. Part of the research in this proposal are the deepening and generalization of some preliminary results published in JCTA, AAM etc. by the applicant and her collaborators, and else are open problems proposed by some well-known international experts in the field. Solving these problems will greatly enrich and improve the enumerative and algebraic combinatorics, and promote the joint development of the combinatorics and some of the relevant disciplines .