Let $st=\{st_1,st_2,...,st_k\}$ be a set of k statistics on permutations with $k\geq 1$. We say that two given subset of permutations $T$ and $T'$ are $st$-Wilf-equivalent if the joint distributions of all statistics in st over the sets of $T$-avoiding permutations $S_n(T)$ and $T'$-avoiding permutations $S_n(T')$ are the same. The main purpose of this paper is the $(cr,nes)$-Wilf-equivalence classes for all single pattern in $S_3$, where cr and nes denote respectively the statistics number of crossings and nestings. One of the main tools that we use is the bijection $\Theta$ from the set of 321-avoiding permutations $S_n(321)$ to the set of 132-avoiding permutations $S_n(132)$ which was exhibited by Elizalde and Pak. They proved that the bijection $\Theta$ preserves the number of fixed points and excedances. Since the given formulation of $\Theta$ is not direct, we show that it can be defined directly by a recursive formula. Then, we prove that it also preserves the number of crossings. These properties of the bijection $\Theta$ leads to an unexpected result related to the q,p-Catalan numbers defined by Randrianarivony.

中文翻译：

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按交叉和嵌套数量细化的受限排列

令 $ st = \ {st_1, st_2, ..., st_k \} $ 是一组 k 个关于 $ k \ geq 1 $ 的排列的统计量。我们说两个给定的置换子集 $T$ 和 $T'$ 是 $st$-Wilf-等价的，如果 st 中所有统计量在 $T$-avoiding permutations$S_n(T)$ 和$ T '$ - 避免排列 $ S_n (T') $ 是一样的。本文的主要目的是对$S_3$中所有单模式的$(cr,nes)$-Wilf-等价类，其中cr和nes分别表示交叉和嵌套的统计数。我们使用的主要工具之一是双射 $\Theta $ 从 321-避免排列的集合 $ S_n (321) $ 到由 Elizalde 和 Pak 展示的 132-避免排列 $ S_n (132) $ 的集合. 他们证明了双射$\Theta$保持不动点数和超越数。由于 $\Theta $ 的给定公式不是直接的，我们证明它可以由递归公式直接定义。然后，我们证明它也保留了交叉的数量。双射 $\Theta$ 的这些属性导致了与 Randrianarivony 定义的 q、p-Catalan 数相关的意外结果。