用混洗方法和队列进行排序
Sorting by shuffling methods and a queue
摘要：
我们研究排序的队列，可以重新排列它们的内容，通过应用排列从一个预定义的集合。这些新的排序设备被称为洗牌队列，我们调查他们对应的定义一些著名的洗牌方法的排列集。如果$\mathbb{Q}_Σ$是对应于洗牌方法$Σ$的洗牌队列，那么我们发现了一些令人惊讶的结果，这些结果与由$\mathbb{Q}_Σ^{\prime}$和$\mathbb{Q}_Σ^{\textsf{pop}}$表示的洗牌队列的两个自然变化有关。这些分别要求在通过每个弹出操作应用或卸载置换之后卸载设备的全部内容。首先，我们证明了通过双端队列进行排序相当于通过可以反转其内容的shuffle队列进行排序。接下来，我们将重点放在按切口排序上。我们证明了可以用$\mathbb{Q}_{\text{cuts}}^{\prime}$排序的排列集合是$321$ -避免可分排列的集合。我们给下限和上限的最大次数的设备必须用于排序排列。此外，我们给出了一个公式的$n$ -排列，$p_{n}(\mathbb{Q}_Σ^{\prime})$，一个可以排序的使用$\mathbb{Q}_Σ^{\prime}$，任何洗牌方法$Σ$，对应于一组不可约排列。我们还表明，$p_{n}(\mathbb{Q}_Σ^{\textsf{pop}})$是由奇数索引斐波那契数$F_{2n-1}$，任何洗牌方法$Σ$具有特定的“后前”属性。剩下的工作是致力于一个令人惊讶的猜想的启发Diaconis和格雷厄姆，其中指出，一个可以排序相同数量的任何给定大小的排列通过使用设备$\mathbb{Q}_{\text{In-sh}}^{\textsf{pop}}$和$\mathbb{Q}_{\text{Monge}}^{\textsf{pop}}$，对应于流行的在洗牌和蒙格洗牌方法。
英文摘要：
We study sorting by queues that can rearrange their content by applying permutations from a predefined set. These new sorting devices are called shuffle queues and we investigate those of them corresponding to sets of permutations defining some well-known shuffling methods. If $\mathbb{Q}_Σ$ is the shuffle queue corresponding to the shuffling method $Σ$, then we find a number of surprising results related to two natural variations of shuffle queues denoted by $\mathbb{Q}_Σ^{\prime}$ and $\mathbb{Q}_Σ^{\textsf{pop}}$. These require the entire content of the device to be unloaded after a permutation is applied or unloaded by each pop operation, respectively. First, we show that sorting by a deque is equivalent to sorting by a shuffle queue that can reverse its content. Next, we focus on sorting by cuts. We prove that the set of permutations that one can sort by using $\mathbb{Q}_{\text{cuts}}^{\prime}$ is the set of the $321$-avoiding separable permutations. We give lower and upper bounds to the maximum number of times the device must be used to sort a permutation. Furthermore, we give a formula for the number of $n$-permutations, $p_{n}(\mathbb{Q}_Σ^{\prime})$, that one can sort by using $\mathbb{Q}_Σ^{\prime}$, for any shuffling method $Σ$, corresponding to a set of irreducible permutations. We also show that $p_{n}(\mathbb{Q}_Σ^{\textsf{pop}})$ is given by the odd indexed Fibonacci numbers $F_{2n-1}$, for any shuffling method $Σ$ having a specific "back-front" property. The rest of the work is dedicated to a surprising conjecture inspired by Diaconis and Graham, which states that one can sort the same number of permutations of any given size by using the devices $\mathbb{Q}_{\text{In-sh}}^{\textsf{pop}}$ and $\mathbb{Q}_{\text{Monge}}^{\textsf{pop}}$, corresponding to the popular In-shuffle and Monge shuffling methods.
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