盒状网格模式的巧合与增长

中文摘要 AI 辅助

盒状网格模式是一种网格模式，其选中的条目位于一个空的轴平行矩形内。我们分类了盒状模式与经典模式和约束模式的巧合，展示了一个真正的双约束巧合，并证明了对于长度至少为五的模式，不存在盒状-双约束巧合。结合已知结果，这表明每个长度至少为五的盒状模式都具有阶乘增长，因此不满足Stanley-Wilf性质。在长度为四时，一个例外轨道由半Baxter数枚举，而剩余的例外轨道$\{2143,3412\}$尚未解决；我们猜想它具有阶乘增长。对于Box(123)，我们推导出一个精确的最大插入恒等式，并证明了次阶乘上界$2^{5n}n^{\beta n}$，其中$\beta=\log_2(2\cos(\pi/7))<0.85$。封闭枚举仍然开放。我们还证明了盒状网格模式的一般一阶矩公式，该公式仅依赖于底层模式的长度；特别地，长度为$n$的均匀随机排列中Box(123)出现次数的期望渐近于$n\log n$。对于Box(12)，我们将出现统计量与强Bruhat序中的上度数联系起来，获得了其最大值、平均值以及分布多项式的精确插入恒等式。我们猜想这些多项式的系数是单峰的。

英文摘要

A boxed mesh pattern is a mesh pattern whose selected entries lie in an empty axis-parallel rectangle. We classify coincidences of boxed patterns with classical and vincular patterns, exhibit a genuinely bivincular coincidence, and prove that no boxed--bivincular coincidence occurs for patterns of length at least five. Together with known results, this shows that every boxed pattern of length at least five has factorial growth and hence fails the Stanley--Wilf property. At length four, one exceptional orbit is enumerated by the semi-Baxter numbers, while the remaining exceptional orbit, $\{2143,3412\}$, is unresolved; we conjecture that it has factorial growth. For Box(123), we derive an exact maximum-insertion identity and prove the subfactorial upper bound $2^{5n}n^{βn}$, where $β=\log_2(2\cos(π/7))<0.85$. A closed enumeration remains open. We also prove a general first-moment formula for boxed mesh patterns that depends only on the length of the underlying pattern; in particular, the expected number of Box(123) occurrences in a uniformly random permutation of length $n$ is asymptotic to $n\log n$. For Box(12), we identify the occurrence statistic with the up-degree in the strong Bruhat order, obtaining its maximum, its mean, and an exact insertion identity for the distribution polynomials. We conjecture that the coefficients of these polynomials are unimodal.

发表机构

• University of Strathclyde（斯特拉斯克莱德大学）
• Nankai University（南开大学）

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