排列中的邻接
Adjacencies in Permutations
摘要：
字母表$ Σ$上的置换是一个序列，其中$ Σ$中的每个元素都只出现一次。给定排列$ π$ =（$π_{1} $，$ π_{2} $，$ π_{3} $，.，$ π_{n} $）在字母表$ Σ$ = $\{ $ 0，1，. . .，n $-$ 1 $\}$如果$ π_{i+1} $ = $ π_{i} $ +1，则在$ π$中的两个连续位置上的元素，例如$ π_{i} $和$ π_{i+1} $，被称为形成\emph{adjacency}。邻接的概念在计算中被广泛使用。$ Σ$上的置换集形成一个对称群，我们称之为P $ _{n} $。单位置换I $ _{n}$ $\in$ P $_{n}$，其中I $_{n}$ =（0，1，2，.，n个$-$ 1）正好有n个$ - $ 1邻接。同样，逆序排列R $_{n} (\in P_{n})$ =（n $-$ 1，n $-$ 2，n $-$ 3，n $-$ 4，.，0)没有邻接关系。我们表示PPDMATHABNE中的置换集，与PPDMATHABNE（k）具有恰好k个邻接。我们研究邻接的变化。一个换位交换相邻的子列表;当其中一个子列表被限制为前缀（后缀）时，则得到一个前缀（后缀）换位。我们称这些操作：换位，前缀换位和后缀换位为块移动。一种特定的邻接类型和一种特定的块移动是密切相关的。在这篇文章中，我们计算的基数P $_{n}$（k），即$ \forall_k \mid $ P $ _{n} $（k）$ \mid $为每种类型的邻接在$O(n^2)$时间。给定一个特定的邻接和相应的块移动，我们表明，$\forall_{k} \mid P_{n}(k)\mid$和预期的移动数排序排列在P $_{n} $是密切相关的。因此，我们提出了一个模型来估计移动的预期数量排序的排列在P $_{n} $块移动。我们展示了前缀换位的结果。由于对称性，这些结果也适用于后缀换位。
英文摘要：
A permutation on an alphabet $ Σ$, is a sequence where every element in $ Σ$ occurs precisely once. Given a permutation $ π$= ($π_{1} $, $ π_{2} $, $ π_{3} $,....., $ π_{n} $) over the alphabet $ Σ$ =$\{ $0, 1, . . . , n$-$1 $\}$ the elements in two consecutive positions in $ π$ e.g. $ π_{i} $ and $ π_{i+1} $ are said to form an \emph{adjacency} if $ π_{i+1} $ =$ π_{i} $+1. The concept of adjacencies is widely used in computation. The set of permutations over $ Σ$ forms a symmetric group, that we call P$ _{n} $. The identity permutation, I$ _{n}$ $\in$ P$_{n}$ where I$_{n}$ =(0,1,2,...,n$-$1) has exactly n$ - $1 adjacencies. Likewise, the reverse order permutation R$_{n} (\in P_{n})$=(n$-$1, n$-$2, n$-$3, n$-$4, ...,0) has no adjacencies. We denote the set of permutations in P$_{n} $ with exactly k adjacencies with P$_{n} $(k). We study variations of adjacency. % A transposition exchanges adjacent sublists; when one of the sublists is restricted to be a prefix (suffix) then one obtains a prefix (suffix) transposition. We call the operations: transpositions, prefix transpositions and suffix transpositions as block-moves. A particular type of adjacency and a particular block-move are closely related. In this article we compute the cardinalities of P$_{n}$(k) i.e. $ \forall_k \mid $P$ _{n} $ (k) $ \mid $ for each type of adjacency in $O(n^2)$ time. Given a particular adjacency and the corresponding block-move, we show that $\forall_{k} \mid P_{n}(k)\mid$ and the expected number of moves to sort a permutation in P$_{n} $ are closely related. Consequently, we propose a model to estimate the expected number of moves to sort a permutation in P$_{n} $ with a block-move. We show the results for prefix transposition. Due to symmetry, these results are also applicable to suffix transposition.
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