论文标题
关于标准Young Tableaux的主要指数的分布
On the distribution of the major index on standard Young tableaux
论文作者
论文摘要
对置换和分区统计数据的研究是列举组合学的经典主题。珀西·麦克马洪（Percy MacMahon）在他的开创性作品中引入了有关排列的主要指数统计数据。在这个扩展的摘要中，我们研究了标准年轻Tableaux的主要指数的众所周知的概括。我们提出了几个新结果。在一个方向上，我们介绍并研究了给定分区形状的标准年轻tableaux的两个部分订单，类似于置换的强大和弱的布鲁哈特订单。这种排名的POSET结构的存在使我们可以针对任意直形形状和某些偏斜形状的标准tableaux进行可实现的主要索引统计数据，并为对称组和Shephard-TODD组具有代表性理论后果。在不同的方向上，我们考虑了主要索引在任意直形状和某些偏斜形状的标准tableaux上的分布。我们根据简单的辅助统计量“ aft”将所有可能的限制定律分类为Canfield-Janson-Janson-Zeilberger，Chen-Wang-Wang等的早期结果。我们还研究单型号，对数洞穴和局部极限属性。
The study of permutation and partition statistics is a classical topic in enumerative combinatorics. The major index statistic on permutations was introduced a century ago by Percy MacMahon in his seminal works. In this extended abstract, we study the well-known generalization of the major index to standard Young tableaux. We present several new results. In one direction, we introduce and study two partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. The existence of such ranked poset structures allows us to classify the realizable major index statistics on standard tableaux of arbitrary straight shape and certain skew shapes, and has representation-theoretic consequences, both for the symmetric group and for Shephard-Todd groups. In a different direction, we consider the distribution of the major index on standard tableaux of arbitrary straight shape and certain skew shapes. We classify all possible limit laws for any sequence of such shapes in terms of a simple auxiliary statistic, "aft," generalizing earlier results of Canfield-Janson-Zeilberger, Chen-Wang-Wang, and others. We also study unimodality, log-concavity, and local limit properties.
