论文标题
看一般的完美造成
A look at generalized perfect shuffles
论文作者
论文摘要
标准的完美散装涉及将$ 2N $卡的甲板分成两个堆栈，并从堆栈中交织。可以通过两种方式进行这种交织的方式，通常称为混乱和散发的混音。 1983年，Diaconis，Graham和Kantor确定了In和Out Shuffles在所有$ N $的$ 2N $卡上产生的置换组。 Diaconis等。结论他们的工作是询问是否可以找到所谓的广义完美混合物的类似结果。对于这些新的散装，我们将$ MN $卡的甲板拆分为$ M $ $堆栈，并以$ m $ shuffle或Out $ m $ shuffle（分别表示$ i_m $ $ $和$ o_m $）将卡与卡片相交。在本文中，我们找到了这两个散装产生的组的结构，用于$ m^k $卡的甲板，以及$ m^y $ shuffles，所有可能的$ m $，$ k $和$ y $的所有可能值。组结构完全由$ k/\ gcd（y，k）$和$ y/\ gcd（y，k）$的均等确定。特别是，组结构独立于$ m $的价值。
Standard perfect shuffles involve splitting a deck of $2n$ cards into two stacks and interlacing the cards from the stacks. There are two ways that this interlacing can be done, commonly referred to as an in shuffle and an out shuffle, respectively. In 1983, Diaconis, Graham, and Kantor determined the permutation group generated by in and out shuffles on a deck of $2n$ cards for all $n$. Diaconis et al. concluded their work by asking whether similar results can be found for so-called generalized perfect shuffles. For these new shuffles, we split a deck of $mn$ cards into $m$ stacks and similarly interlace the cards with an in $m$-shuffle or out $m$-shuffle (denoted $I_m$ and $O_m$, respectively). In this paper, we find the structure of the group generated by these two shuffles for a deck of $m^k$ cards, together with $m^y$-shuffles, for all possible values of $m$, $k$, and $y$. The group structure is completely determined by $k/\gcd(y,k)$ and the parity of $y/\gcd(y,k)$. In particular, the group structure is independent of the value of $m$.
