标题： 避免部分有序广义模式$k$-$σ$-$k$的出现 显示英文标题

标题： Avoidance of Partially Ordered Generalized Patterns of the form $k$-$σ$-$k$

摘要： Sergey Kitaev 已经证明，避免广义模式$\sigma$-$k$的排列的指数生成函数，其中$\sigma$是一个没有破折号的模式，$k$是比$\sigma$中最大元素大 1 的模式，由避免$\sigma$的排列的指数生成函数决定。 我们证明对于避免所有广义模式$\sigma_1$-$k_1$, $...$, $\sigma_n$-$k_n$的排列也成立，其中$\sigma_1$, $...$, $\sigma_n$是没有连字符的模式，而$k_i$是比$\sigma_i$中最大的元素大一。 同样，避免部分有序广义模式$k_1$-$\sigma_1$-$k_1$，$...$，$k_n$-$\sigma_n$-$k_n$的排列的指数生成函数可以从避免广义模式$\sigma_1$，$...$，$\sigma_n$的排列的指数生成函数确定，其中$\sigma_1$，$...$，$\sigma_n$是没有连字符的模式，而$k_i$是比$\sigma_i$中最大元素大 1 的数。 利用这一点，我们构造了双色集合划分与避免部分有序广义模式 3-12-3（即避免模式 3-12-4 和 4-12-3）的排列之间的双射。 通过两次使用这种方法，我们找到了避免部分有序广义模式 3-121-3 的排列的指数生成函数的闭式公式。 最后，我们给出了单个部分有序广义模式何时具有相同避免者的完整分类。 显示英文摘要

摘要： Sergey Kitaev has shown that the exponential generating function for permutations avoiding the generalized pattern $\sigma$-$k$, where $\sigma$ is a pattern without dashes and $k$ is one greater than the biggest element in $\sigma$, is determined by the exponential generating function for permutations avoiding $\sigma$. We show that this also holds for permutations avoiding all the generalized patterns $\sigma_1$-$k_1$, $...$, $\sigma_n$-$k_n$, where $\sigma_1$, $...$, $\sigma_n$ are patterns without dashes and $k_i$ is one greater than the biggest element in $\sigma_i$. Similarly the exponential generating function for permutations avoiding the partially ordered generalized patterns $k_1$-$\sigma_1$-$k_1$, $...$, $k_n$-$\sigma_n$-$k_n$ can be determined from the exponential generating function for permutations avoiding the generalized patterns $\sigma_1$, $...$, $\sigma_n$, where $\sigma_1$, $...$, $\sigma_n$ are patterns without dashes and $k_i$ is one greater than the largest element in $\sigma_i$. Using this we construct a bijection between bicolored set partitions and permutations avoiding the partially ordered generalized pattern 3-12-3 (that is, permutations avoiding both the patterns 3-12-4 and 4-12-3). By using this method twice, we find a closed formula for the exponential generating function for permutations avoiding the partially ordered generalized pattern 3-121-3. Finally, we give a complete classification of when single partially ordered generalized patterns have the same set of avoiders.