论文标题
部分可观测时空混沌系统的无模型预测
Integer Representations of the Generalized Symmetric Groups
论文作者
论文摘要
在本文中，我们在广义对称组$ g（m，1，n）$上构建了一个混合基数系统，该系统是一个复杂的反射组，具有$ b_n^{（m）} $的root系统。我们还在集合$ \ {1，\ cdots，m^nn！\} $与$ g（m，1，n）$的元素的所有正整数之间建立一对一的对应关系，通过与该组相关的次级函数。此外，我们通过在$ g（m，1，n）$上定义反演统计量来为$ g（m，1，n）$提供新的枚举系统。最后，我们证明\ textIt {flag-major index}在$ g（m，1，n）$上与此反转统计量进行了等式。因此，相对于长度函数$ l $，Flag-Major索引是$ g（m，1，n）$的Mahonian。
In this paper, we construct a mixed-base number system over the generalized symmetric group $G(m,1,n)$, which is a complex reflection group with a root system of type $B_n^{(m)}$. We also establish one-to-one correspondence between all positive integers in the set $\{1,\cdots,m^nn!\}$ and the elements of $G(m,1,n)$ by constructing the subexceedant function in relation to this group. In addition, we provide a new enumeration system for $G(m,1,n)$ by defining the inversion statistic on $G(m,1,n)$. Finally, we prove that the \textit{flag-major index} is equi-distributed with this inversion statistic on $G(m,1,n)$. Therefore, the flag-major index is Mahonian on $G(m,1,n)$ with respect to the length function $L$.
