用Garsia-Gessel双射计数圈积
Enumerating Wreath Products Via Garsia-Gessel Bijections
摘要：
本文推广了Garsia和Gessel的两个双射，计算了对称群与循环群的圈积上两个向量统计量$(\des_G, \maj,\ell_G, \col)$和$(\des_G, \ides_G, \maj, \imaj, \col, \icol)$的生成函数.这里，$\des_G$、$\ell_G$、$\maj$、$\col$、$\ides_G$、$\imaj_G$和$\icol$分别表示下降数、长度、主索引、颜色权重、逆下降、逆主索引和逆颜色权重。我们的主要公式推广和统一了几个已知的身份由于Brenti，Carlitz，Chow-Gessel，Garsia-Gessel和Reiner的各种分布的统计数据的Coxeter群的类型$A$和$B$。
英文摘要：
We generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics $(\des_G, \maj,\ell_G, \col)$ and $(\des_G, \ides_G, \maj, \imaj, \col, \icol)$ over the wreath product of a symmetric group by a cyclic group. Here $\des_G$, $\ell_G$, $\maj$, $\col$, $\ides_G$, $\imaj_G$, and $\icol$ denote the number of descents, length, major index, color weight, inverse descents, inverse major index, and inverse color weight, respectively. Our main formulas generalize and unify several known identities due to Brenti, Carlitz, Chow-Gessel, Garsia-Gessel, and Reiner on various distributions of statistics over Coxeter groups of type $A$ and $B$.
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。
