广义置换上的模式避免

模式避免问题是当前计数组合的研究热点之一。置换与重集的置换上模式避免的研究日趋完备，广义置换是置换与重集的置换的统一与推广，其上的相关研究才刚开始。前期我们在广义置换上引入了模式避免的概念，借助Young表理论将Knuth在置换上的经典结果延伸到了广义置换，即3长模式在广义置换中形成一个Wilf-等价类。在此基础上，本项目将从Wilf-等价类的划分与相关计数、模式避免与Young表、统计量三个方面进一步研究广义置换上模式避免问题。主要研究内容包括：系统研究3长模式集在广义置换上Wilf-等价类的划分与相关计数；研究Kostka数的对数凹性、Schur凹性及极值结构，进一步挖掘Young表理论在模式避免问题中的应用；探索广义置换上逆序、下降数、主指标等统计量的定义和生成函数，研究广义置换关于其型的生成函数的对称性。本项目的开展将对统一和完善置换与重集置换上模式避免的研究有着重要意义。

Today the study of pattern avoidance in permutations and words is a very active field. In the last 30 years or so, the area of pattern avoidance in set and multiset permutations has grown dramatically, resulting in many hundreds of published papers . Extending the concept of pattern avoidance in set and multiset permutations, we have defined the pattern avoidance on genenralized permutations and shown that there is only one Wilf-equivalent class of 3 length pattern. Based on this result, this project will make a further study of pattern avoidance of generalized permutations in three ways, via classification of Wilf-equivalent classes, pattern avoidance and Young tableaux, and statistics. More precisely, we firstly will complete the classification of Wilf-equivalent classes for pairs and triples of 3-patterns in generalized permutations and give the relative enumerations. Secondly, we will show that Kostka number is log-concave, study that whether the skew Kostka number is Schur-concave, and find the maximal of a kind of Kostka number. Meanwhile, we will search more applications of Young tableaux in pattern avoidance. Finally, we focus our attention on statistics of generalized permutations. In this part, we mainly study the generating functions of inversions, descents, major, and the type of generalized permutations. In a word, we think the project will make a difference in the study of pattern avoidance.