带色错位上的符号马霍尼多项式
Signed Mahonian Polynomials on Colored Derangements
AI总结：
研究带色错位上的符号马霍尼多项式，推导偶长度彩色错位计数公式及偶奇长度错位数量差公式。
摘要：
在经典的Weyl群W上，主要指数的多项式$\sum_{π∈W}q^{maj(π)}$被称为马霍尼多项式，而同时包含符号的主要指数多项式$\sum_{π∈W}(-1)^{l(π)}q^{maj(π)}$被称为W上的符号马霍尼多项式，其中l是通过生成集S定义的长度函数。本文研究的是在彩色错位集合$D_{n}^{(c)}$上的符号马霍尼多项式$$\sum_{π∈D_{n}^{(c)}}(-1)^{L(π)}q^{fmaj(π)}$$，其中L是通过Bremke和Malle在$G_{c,n}$中描述的复根系定义的长度函数，而fmaj是由Adin和Roichman在$G_{c,n}$中定义的旗-主要指数，这是一个马霍尼统计量。作为在彩色错位集合上符号马霍尼多项式公式的应用，我们将推导出当c为偶数时，在$G_{c,n}$中计算偶长度彩色错位的公式。最后，我们通过提供一个公式，来计算在$G_{c,n}$中偶长度和奇长度错位的数量之差，无论c是奇数还是偶数。
英文摘要：
The polynomial $\sum_{π\in W}q^{maj(π)}$ of major index over a classical Weyl group $W$ with a generating set $S$ is called the Mahonian polynomial over $W$, and also the polynomial $\sum_{π\in W}(-1)^{l(π)}q^{maj(π)}$ of major index together with sign over the group $W$ is called the signed Mahonian polynomial over the group $W$, where $l$ is the length function on $W$ defined in terms of the generating set $S$. We concern with the signed Mahonian polynomial $$\sum_{π\in D_{n}^{(c)}}(-1)^{L(π)}q^{fmaj(π)}$$ on the set $D_{n}^{(c)}$ of colored derangements in the group $G_{c,n}$ of colored permutations, where $L$ denotes the length function defined by means of a complex root system described by Bremke and Malle in $G_{c,n}$ and $fmaj$ defined by Adin and Roichman in $G_{c,n}$ represents the \textit{flag-major index}, which is a Mahonian statistic. As an application of the formula for signed Mahonian polynomials on the set of colored derangements, we will derive a formula to count colored derangements of even length in $G_{c,n}$ when $c$ is an even number. Finally, we conclude by providing a formula for the difference between the number of derangements of even and odd lengths in $G_{c,n}$ for every positive integer $c$, regardless of whether c is odd or even.
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