A permutation $\pi: [n] \rightarrow [n]$ is a Baxter permutation if and only if it does not contain either of the patterns $2-41-3$ and $3-14-2$. Baxter permutations are one of the most widely studied subclasses of general permutation due to their connections with various combinatorial objects such as plane bipolar orientations and mosaic floorplans, etc. In this paper, we introduce a novel succinct representation (i.e., using $o(n)$ additional bits from their information-theoretical lower bounds) for Baxter permutations of size $n$ that supports $\pi(i)$ and $\pi^{-1}(j)$ queries for any $i \in [n]$ in $O(f_1(n))$ and $O(f_2(n))$ time, respectively. Here, $f_1(n)$ and $f_2(n)$ are arbitrary increasing functions that satisfy the conditions $\omega(\log n)$ and $\omega(\log^2 n)$, respectively. This stands out as the first succinct representation with sub-linear worst-case query times for Baxter permutations. Additionally, we consider a subclass of Baxter permutations called \textit{separable permutations}, which do not contain either of the patterns $2-4-1-3$ and $3-1-4-2$. In this paper, we provide the first succinct representation of the separable permutation $\rho: [n] \rightarrow [n]$ of size $n$ that supports both $\rho(i)$ and $\rho^{-1}(j)$ queries in $O(1)$ time. In particular, this result circumvents Golynski's [SODA 2009] lower bound result for trade-offs between redundancy and $\rho(i)$ and $\rho^{-1}(j)$ queries. Moreover, as applications of these permutations with the queries, we also introduce the first succinct representations for mosaic/slicing floorplans, and plane bipolar orientations, which can further support specific navigational queries on them efficiently.

翻译：若置换 $\pi: [n] \rightarrow [n]$ 不包含模式 $2-41-3$ 与 $3-14-2$ 中的任意一个，则称其为Baxter置换。由于Baxter置换与平面双极定向、马赛克平面布局等多种组合对象存在关联，它已成为一般置换中被研究最广泛的子类之一。本文针对规模为 $n$ 的Baxter置换，提出了一种新颖的简洁表示（即仅使用信息论下界之外的 $o(n)$ 额外比特），该表示可在 $O(f_1(n))$ 与 $O(f_2(n))$ 时间内分别支持对任意 $i \in [n]$ 的 $\pi(i)$ 与 $\pi^{-1}(j)$ 查询。其中 $f_1(n)$ 与 $f_2(n)$ 为满足条件 $\omega(\log n)$ 与 $\omega(\log^2 n)$ 的任意递增函数。这是首个对Baxter置换实现最坏情况下亚线性查询时间的简洁表示。此外，本文还考察了Baxter置换的一个子类——\textit{可分离置换}，该类置换不包含模式 $2-4-1-3$ 与 $3-1-4-2$ 中的任意一个。我们首次给出了规模为 $n$ 的可分离置换 $\rho: [n] \rightarrow [n]$ 的简洁表示，该表示能以 $O(1)$ 时间复杂度同时支持 $\rho(i)$ 与 $\rho^{-1}(j)$ 查询。特别地，该结果突破了Golynski [SODA 2009] 关于冗余度与 $\rho(i)$、$\rho^{-1}(j)$ 查询之间权衡的下界结论。进一步地，基于这些置换及其查询操作的应用，我们还首次提出了针对马赛克/切片平面布局及平面双极定向的简洁表示，这些表示能高效支持对其进行的特定导航查询。