关于带对称阴影的网格模式123与132的（联合）等分布
On (joint) equidistributions of mesh patterns 123 and 132 with symmetric shadings
AI总结：
针对排列模式中网格模式123与132的分布问题，本文通过双射证明确立了75种带对称阴影的等分布对及36种非对称等分布，延续了短长度网格模式分布的系统研究。
摘要：
排列模式领域中一个自1973年起就在文献中受到广泛关注的著名问题，是寻找一个双射证明，以论证避免123的排列与避免132的排列数量相等，二者均由卡特兰数计数。尽管存在这种数量等价性，但模式123与132的出现分布是不同的。当将123和132视为网格模式并选择性地对方格着色时，会出现类似的情形，即便此时的避免性由贝尔数或其他序列而非卡特兰数定义。\n然而，计算机实验表明网格模式123与132实际上可能是等分布的。此外，通过考虑相对于反对角线的对称阴影，最多可能存在93个这样的等分布对。本文确立了75种此类等分布，将剩余情形的证明留作开放问题。作为副产品，我们还证明了36个相关的非对称等分布。我们所有的证明都是双射性的，涉及交换所讨论模式的出现，从而证明了它们的联合等分布。我们的研究结果是对Kitaev与Zhang在2019年开启的短长度网格模式分布系统研究的延续。
英文摘要：
A notable problem within permutation patterns that has attracted considerable attention in literature since 1973 is the search for a bijective proof demonstrating that 123-avoiding and 132-avoiding permutations are equinumerous, both counted by the Catalan numbers. Despite this equivalence, the distributions of occurrences of the patterns 123 and 132 are distinct. When considering 123 and 132 as mesh patterns and selectively shading boxes, similar scenarios arise, even when avoidance is defined by the Bell numbers or other sequences, rather than the Catalan numbers. However, computer experiments suggest that mesh patterns 123 and 132 may indeed be equidistributed. Furthermore, by considering symmetric shadings relative to the anti-diagonal, a maximum of 93 such equidistributed pairs can potentially exist. This paper establishes 75 such equidistributions, leaving the justification of the remaining cases as open problems. As a by-product, we also prove 36 relevant non-symmetric equidistributions. All our proofs are bijective and involve swapping occurrences of the patterns in question, thereby demonstrating their joint equidistribution. Our findings are a continuation of the systematic study of distributions of short-length mesh patterns initiated by Kitaev and Zhang in 2019.
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。
