Labeled Partitions and the q-Derangement Numbers
Labeled Partitions and the q-Derangement Numbers
复制标题
DOI：
10.1137/06066326x
复制
发表时间：
2006-06
期刊：
影响因子：
--
通讯作者：
William Y. C. Chen;Deheng Xu
中科院分区：
文献类型：
--
作者：
William Y. C. Chen;Deheng Xu
受MacMahon关于排列上主指数分布的著名定理的原始证明的启发，我们给出了他的论点在标记分区方面的一个重新表述。在此框架下，我们建立了标记分区的分解定理，该定理类似于置换分解为无序点和不动点。这种分解意味着对Wachs关于无序部分的公式和排列的主要指数的重新表述，该公式是为了客观地处理Gessel关于$q$-无序数的公式而导出的。
Inspired by MacMahon's original proof of his celebrated theorem on the distribution of the major index over permutations, we give a reformulation of his argument in terms of labeled partitions. In this framework, we establish a decomposition theorem for labeled partitions which is analogous to the decomposition of a permutation into derangement points and fixed points. This decomposition implies a reformulation of Wachs's formula concerning the derangement parts and the major index on permutations which was derived in order to present a bijective treatment of Gessel's formula on the $q$-derangement numbers.