发表机构
University of Ljubljana, Faculty of Mathematics and Physics; Institute for Mathematics, Physics and Mechanics; KTH Royal Institute of Technology（卢布尔雅那大学数学物理学院; 数学、物理与力学研究所; 瑞典皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为交替符号矩形引入扩展模式避免，对长度三的四个模式建立递推枚举系统，并给出312的猜想证明及213到小Schröder路径的双射。
AI 中文摘要
我们引入了交替符号矩形（ASRs）的扩展模式避免，这是交替符号矩阵（ASMs）的自然矩形推广。如果一个ASR是某个避免模式$\pi$的ASM的左上角，则称该ASR扩展地避免模式$\pi$。对于等价类$\{312, 132, 213, 231\}$中的四个长度为三的模式，我们建立了完整的递推关系系统，用于枚举大小为$r \times k$、具有指定非空行数$d$的扩展避免$\pi$的ASRs。对于$\pi = 312$，我们进一步猜想了一个闭式表达式，并通过涉及Schröder ballot数、精细Schröder数和不垂直穿过主对角线的Delannoy路径的双射，在几个对角切片上证明了该猜想。对于大小为$(r-1) \times (r+1)$、扩展避免$213$的ASRs，我们给出了到长度为$r$的小Schröder路径的双射。剩余的长度为三的模式$123$和$321$则更加难以捉摸，这反映了ASMs的情况。
英文摘要
We introduce extendable pattern avoidance for alternating sign rectangles (ASRs), the natural rectangular generalization of alternating sign matrices (ASMs). An ASR extendably avoids a pattern $π$ if it is the upper left corner of an ASM avoiding $π$. For each of the four length-three patterns in the equivalence class $\{312, 132, 213, 231\}$ we establish a complete system of recurrence relations enumerating extendably $π$-avoiding ASRs of size $r \times k$ with a prescribed number $d$ of nonempty rows. For $π= 312$ we further conjecture a closed-form expression and prove it on several diagonal slices via bijections involving Schröder ballot numbers, refined Schröder numbers and Delannoy paths that do not cross the main diagonal vertically. For ASRs of size $(r-1) \times (r+1)$ extendably avoiding $213$ we give a bijection to little Schröder paths of length $r$. The remaining patterns of length three, $123$ and $321$, are more elusive, mirroring the situation of ASMs.
Comments25 pages
