发表机构
Ramapo College of New Jersey（新泽西拉马波学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究避免123模式且相邻元素差不超过m的排列，通过有限状态转移矩阵精确枚举，证明增长率小于4并收敛于Catalan常数，同时揭示该约束打破Wilf等价性。
AI 中文摘要
我们研究排列 $\pi \in S_n$，它们避免 $123$ 模式，同时满足邻接约束 $\lvert \pi_{i+1}-\pi_i\rvert \le m$，其中 $m\in\mathbb{Z}^{+}$。我们首先证明每个可容许排列都局限在反向对角线的垂直距离 $m$ 之内，这揭示了模式避免与有界邻接之间相互作用所施加的强全局限制。对于每个固定的 $m$，我们随后构造该族的一个精确有限状态描述，具有恰好 $2^{m+1}-m-1$ 个可实现状态。这产生了一个显式的转移矩阵枚举，并且特别地，对于每个固定的 $m$ 得到一个有理生成函数；因此，枚举序列满足最终常系数线性递推。我们进一步将指数增长率与相应转移矩阵的谱半径 $\rho_m$ 相关联，并证明 $\rho_m<4$，关于 $m$ 非递减，且当 $m\to\infty$ 时收敛到 Catalan 增长常数 $4$。最后，我们证明有界邻接约束打破了普通长度三的 Wilf 等价性，并发展了该族的算法和图论解释，包括与路径图的幂中的约束哈密顿路径的对应关系。
英文摘要
We study permutations $π\in S_n$ that avoid $123$ while satisfying the adjacency constraint $\lvert π_{i+1}-π_i\rvert \le m$ for some $m\in\mathbb{Z}^{+}$. We first show that every admissible permutation is localized within vertical distance $m$ of the reverse diagonal, revealing a strong global restriction imposed by the interaction between pattern avoidance and bounded adjacency. For every fixed $m$, we then construct an exact finite-state description of the family, with precisely $2^{m+1}-m-1$ realizable states. This yields an explicit transfer-matrix enumeration and, in particular, a rational generating function for every fixed $m$; consequently, the enumeration sequence satisfies an eventual constant-coefficient linear recurrence. We further identify the exponential growth rate with the spectral radius $ρ_m$ of the corresponding transfer matrix and show that $ρ_m<4$, is nondecreasing in $m$, and converges to the Catalan growth constant $4$ as $m\to\infty$. Finally, we show that the bounded-adjacency constraint breaks the ordinary length-three Wilf equivalence, and we develop algorithmic and graph-theoretic interpretations of the family, including a correspondence with constrained Hamiltonian paths in powers of the path graph.
Comments49 pages
