In this paper, we study the graph induced by the $\textit{2-swap}$ permutation on words with a fixed Parikh vector. A $2$-swap is defined as a pair of positions $s = (i, j)$ where the word $w$ induced by the swap $s$ on $v$ is $v[1] v[2] \dots v[i - 1] v[j] v[i+1] \dots v[j - 1] v[i] v[j + 1] \dots v[n]$. With these permutations, we define the $\textit{Configuration Graph}$, $G(P)$ defined over a given Parikh vector. Each vertex in $G(P)$ corresponds to a unique word with the Parikh vector $P$, with an edge between any pair of words $v$ and $w$ if there exists a swap $s$ such that $v \circ s = w$. We provide several key combinatorial properties of this graph, including the exact diameter of this graph, the clique number of the graph, and the relationships between subgraphs within this graph. Additionally, we show that for every vertex in the graph, there exists a Hamiltonian path starting at this vertex. Finally, we provide an algorithm enumerating these paths from a given input word of length $n$ with a delay of at most $O(\log n)$ between outputting edges, requiring $O(n \log n)$ preprocessing.

翻译：本文研究固定帕里克向量（Parikh vector）的单词上由$\textit{2-交换}$置换所诱导的图。2-交换定义为一对位置$s = (i, j)$，其中通过交换$s$作用于$v$所得到的单词$w$为$v[1] v[2] \dots v[i - 1] v[j] v[i+1] \dots v[j - 1] v[i] v[j + 1] \dots v[n]$。利用这些置换，我们定义了基于给定帕里克向量的$\textit{配置图}$ $G(P)$。$G(P)$中的每个顶点对应一个具有帕里克向量$P$的唯一单词，且任意两个单词$v$和$w$之间存在一条边当且仅当存在一个交换$s$使得$v \circ s = w$。我们给出了该图的若干关键组合性质，包括图的精确直径、团数以及子图之间的关联关系。此外，我们证明了图中每个顶点均存在一条起始于该顶点的哈密顿路径。最后，我们提出了一种枚举算法，该算法从给定长度为$n$的输入单词出发，在输出边之间至多存在$O(\log n)$的延迟，且需要$O(n \log n)$的预处理时间。