关于$132$和$123$的阶梯编码和边界网格的两个猜想的解决--避免置换
Resolving Two Conjectures on Staircase Encodings and Boundary Grids of $132$ and $123$-avoiding permutations
摘要：
本文分析了某些置换的模式避免与阶梯网格和边界网格上的图之间的关系，并证明了Bean，Tannock和Ulfarsson（2015）提出的两个定理。更具体地说，本文列举了一类阶梯编码，并证明了建立在边界网格上的downcore图是纯的当且仅当对应于边界网格的置换避开了经典模式123和2143.
英文摘要：
This paper analyzes relations between pattern avoidance of certain permutations and graphs on staircase grids and boundary grids, and proves two conjectures posed by Bean, Tannock, and Ulfarsson (2015). More specifically, this paper enumerates a certain family of staircase encodings and proves that the downcore graph, a certain graph established on the boundary grid, is pure if and only if the permutation corresponding to the boundary grid avoids the classical patterns 123 and 2143.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
