B型$q$-Mahonian数的组合性质与对数凹性
Combinatorics of $q$-Mahonian numbers of type $B$ and log-concavity
AI总结：
本文研究B型Mahonian数及其$q$-模拟，给出公式、生成函数与格路径、分拆和铺砌解释，并证明其强$q$-对数凹性及由此得到的对数凹性和单峰性。
摘要：
本文是Arslan早期工作的延续；他通过在超八面体群$B_n$上使用一种新统计量引入了B型Mahonian数，以回应其发表于arXiv:2404.05099v1、题为《B型Mahonian数的组合解释》的论文中提出的问题。我们首先给出B型Mahonian数主对角线上或其下方次对角线的Knuth-Netto公式和生成函数，随后通过格路径/分拆以及铺砌给出其组合解释。接着，我们利用在超八面体群$B_n$排列上引入的一种新统计量，提出B型Mahonian数的$q$-模拟，然后研究其基本性质，以及通过格路径/分拆和铺砌得到的组合解释。最后，我们以组合方式证明，B型Mahonian数的$q$-模拟构成关于$k$的强$q$-对数凹多项式序列，这意味着B型Mahonian数构成关于$k$的对数凹序列，并因此是单峰的。
英文摘要：
This paper is a continuation of earlier work of Arslan \cite{Ars}, who introduced the Mahonian number of type $B$ by using a new statistic on the hyperoctahedral group $B_{n}$, in response to questions he suggested in his paper entitled "{\it A combinatorial interpretation of Mahonian numbers of type $B$}" published in arXiv:2404.05099v1. We first give the Knuth-Netto formula and generating function for the subdiagonals on or below the main diagonal of the Mahonian numbers of type $B$, then its combinatorial interpretations by lattice path/partition and tiling. Next, we propose a $q$-analogue of Mahonian numbers of type $B$ by using a new statistics on the permutations of the hyperoctahedral group $B_n$ that we introduced, then we study their basic properties and their combinatorial interpretations by lattice path/partition and tiling. Finally, we prove combinatorially that the $q$-analogue of Mahonian numbers of type $B$ form a strongly $q$-log-concave sequence of polynomials in $k$, which implies that the Mahonian numbers of type $B$ form a log-concave sequence in $k$ and therefore unimodal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
