符号Euler-Mahonian恒等式
Signed Euler-Mahonian identities
摘要：
1992年Désarmenien和Foata给出了$\mathfrak{S}_n$上的带符号欧拉多项式和经典欧拉多项式之间的关系，同年Wachs提出了一个改进的版本，称为带符号欧拉-Mahonian恒等式，并给出了一个双射证明。通过推广这个双射，本文将上述结果推广到$B_n$，$D_n$和复反射群$G(r,1,n)$的Coxeter群，其中“符号”取为任何一维特征标.一些获得的恒等式可以进一步限制在一些特定的排列集合上。我们还得到了$B_n$和$D_n$型的一些新的有趣的符号平衡多项式。
英文摘要：
A relationship between signed Eulerian polynomials and the classical Eulerian polynomials on $\mathfrak{S}_n$ was given by Désarménien and Foata in 1992, and a refined version, called signed Euler-Mahonian identity, together with a bijective proof were proposed by Wachs in the same year. By generalizing this bijection, in this paper we extend the above results to the Coxeter groups of types $B_n$, $D_n$, and the complex reflection group $G(r,1,n)$, where the `sign' is taken to be any one-dimensional character. Some obtained identities can be further restricted on some particular set of permutations. We also derive some new interesting sign-balance polynomials for types $B_n$ and $D_n$.
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