For a permutation $\pi: [K]\rightarrow [K]$, a sequence $f: \{1,2,\cdots, n\}\rightarrow \mathbb R$ contains a $\pi$-pattern of size $K$, if there is a sequence of indices $(i_1, i_2, \cdots, i_K)$ ($i_1<i_2<\cdots<i_K$), satisfying that $f(i_a)<f(i_b)$ if $\pi(a)<\pi(b)$, for $a,b\in [K]$. Otherwise, $f$ is referred to as $\pi$-free. For the special case where $\pi = (1,2,\cdots, K)$, it is referred to as the monotone pattern. \cite{newman2017testing} initiated the study of testing $\pi$-freeness with one-sided error. They focused on two specific problems, testing the monotone permutations and the $(1,3,2)$ permutation. For the problem of testing monotone permutation $(1,2,\cdots,K)$, \cite{ben2019finding} improved the $(\log n)^{O(K^2)}$ non-adaptive query complexity of \cite{newman2017testing} to $O((\log n)^{\lfloor \log_{2} K\rfloor})$. Further, \cite{ben2019optimal} proposed an adaptive algorithm with $O(\log n)$ query complexity. However, no progress has yet been made on the problem of testing $(1,3,2)$-freeness. In this work, we present an adaptive algorithm for testing $(1,3,2)$-freeness. The query complexity of our algorithm is $O(\epsilon^{-2}\log^4 n)$, which significantly improves over the $O(\epsilon^{-7}\log^{26}n)$-query adaptive algorithm of \cite{newman2017testing}. This improvement is mainly achieved by the proposal of a new structure embedded in the patterns.

翻译：对于置换$\pi: [K]\rightarrow [K]$，若存在下标序列$(i_1, i_2, \cdots, i_K)$（满足$i_1<i_2<\cdots<i_K$），使得对任意$a,b\in [K]$，当$\pi(a)<\pi(b)$时均有$f(i_a)<f(i_b)$，则称序列$f: \{1,2,\cdots, n\}\rightarrow \mathbb R$包含大小为$K$的$\pi$-模式。否则称$f$为$\pi$-自由。特殊情形$\pi = (1,2,\cdots, K$对应单调模式。\cite{newman2017testing}首次提出单侧错误检测$\pi$-自由性问题，重点研究了单调置换与$(1,3,2)$置换两类问题。对于单调置换$(1,2,\cdots,K)$的检测问题，\cite{ben2019finding}将\cite{newman2017testing}中$(\log n)^{O(K^2)}$的非自适应查询复杂度改进为$O((\log n)^{\lfloor \log_{2} K\rfloor})$，随后\cite{ben2019optimal}提出查询复杂度为$O(\log n)$的自适应算法。然而，$(1,3,2)$-自由性检测问题尚未取得进展。本文提出检测$(1,3,2)$-自由性的自适应算法，其查询复杂度为$O(\epsilon^{-2}\log^4 n)$，较\cite{newman2017testing}中$O(\epsilon^{-7}\log^{26}n)$查询的自适应算法实现显著提升。该改进主要归功于模式中嵌入新结构的提出。