Haglund猜想的推广与k-Stirling排列统计量的研究

近年来，排列统计量理论的一个重要进展是著名组合学家Remmel和Wilson于2015年证明了Haglund猜想，该猜想是排列统计量理论的核心定理——MacMahon等分布定理的深刻推广。本项目拟借助k-Stirling排列理论对这个重要结果进行推广。k-Stirling排列与许多组合结构（尤其是递增树、格路、匹配、逆序序列）有密切的联系，受到了许多著名组合学家（包括R.P.Stanley，I.M.Gessel，M.Bóna，J.Haglund以及陈永川院士等）的极大关注。本项目的研究内容主要有两个方面，一是利用k-Stirling排列理论推广Remmel和Wilson证明的Haglund猜想，二是建立k-Stirling排列的各种统计量之间的深刻联系。

In 2015, Remmel and Wilson proved Haglund's conjecture, which is a deep extension of MacMahon's equidistribution theorem. In this project, we will use the theory of k-Stirling permutations to give an extension of this important result. k-Stirling permutations have received quite some attention of many famous combinatorial scientists (including R.P.Stanley, I.M.Gessel, M.Bóna, J.Haglund and William Y.C.Chen), due to relations to other structures such as increasing trees, lattice paths, matchings, inversion sequences. In this project we will not only give an extension of the result of Remmel and Wilson, but also establish the deep relationships among the statistics of k-Stirling permutations.