发表机构
University of Haifa; Makerere University; University of Tennessee（海法大学; Makerere大学; 田纳西大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究由集合划分平化得到的排列集$\uD835\uDCA9_n$中元素对长度为4的模式的避免问题，证明7种模式下避免数为卡特兰数，3种模式下为卡特兰数的二项变换，通过细化计数序列结合生成函数求解完成推导。
AI 中文摘要
设$\uD835\uDCA9_n$表示由集合$[n] = \{1,\ldots,n\}$的划分经过平化过程得到的长度为$n$的不同排列构成的集合。本文研究$\uD835\uDCA9_n$中元素避免单个长度为4的经典模式的问题。设$p_n(\tau)$表示$\uD835\uDCA9_n$中避免模式$\tau$的元素个数。我们证明，对于7个长度为4的模式，当$n \geq 1$时，$p_n(\tau) = C_{n-1}$，这为卡特兰数序列提供了新的组合解释。此外，我们证明，对于另外3个模式，$p_n(\tau)$对应卡特兰数的二项变换。为建立这些结果，我们在每种情况下适当细化计数序列$p_n(\tau)$，以得到对应生成函数满足的函数方程组，随后可显式求解这些函数方程组，进而确定每种情况下的$p_n(\tau)$。
英文摘要
Let $\mathcal{P}_n$ denote the set of distinct permutations of length $n$ that arise from the flattening process applied to the partitions of $[n]=\{1,\ldots,n\}$. In this paper, we consider the problem of avoidance of a single classical pattern of length four by members of $\mathcal{P}_n$. Let $p_n(τ)$ denote the number of members of $\mathcal{P}_n$ that avoid the pattern $τ$. We show that $p_n(τ)=C_{n-1}$ for all $n \geq 1$ for seven patterns of length four yielding new combinatorial interpretations of the Catalan number sequence. Further, we show that $p_n(τ)$ corresponds to the binomial transform of Catalan numbers for three other patterns. To establish our results, we suitably refine the counting sequence $p_n(τ)$ in each case so as to obtain a system of functional equations satisfied by the corresponding generating functions. These functional equations may then be solved explicitly leading to a determination of $p_n(τ)$ in each case.
