AI 中文总结
该研究从两方面研究Canon排列的模式避免，解决Laudone的三个猜想，确定字母表大小为3时的相关计数，分类禁集并推导生成函数相关结果，提出问题与猜想。
AI 中文摘要
我们从两个互补的角度研究Canon排列中的经典模式避免问题。首先，对于任意字母表大小，我们通过给出规范结构分解解决了Laudone的三个猜想。其次，我们将字母表大小固定为3，确定了对任意Λ⊆𝔖₃的c₃ᵏ(Λ)。重标记归约和六类格路定理表明，每个固定底层排列分量的基数为0、1、Cₖ、Bₖ、Qₖ或Tₖ，其中Cₖ表示第k个卡特兰数，Tₖ=f⁽ᵏ,ᵏ,ᵏ⁾是无限制矩形表数，Bₖ计数每行都满足1在3之前的三行格路，Qₖ计数避免321的三行格路。64个禁集产生12个计数公式，我们对其下降多项式一致为回文或γ-正的禁集进行分类，并为所有受限γ-正类给出初等阿贝尔2群的布尔切换作用。最后，我们推导了Bₖ生成函数的闭式公式和三次代数方程、Qₖ的精确有限多重和，并提出了若干问题与猜想。
英文摘要
We study avoidance of patterns of length $3$ with three distinct letters in canon permutations. We reduce the problem to studying pattern avoidance in lattice words and show that there are $6$ such pattern avoiding classes. This shows that there are $12$ classes for the original Canon permutation pattern avoidance problem. We also give descent refinements for these classes and classify the patterns for which the descent enumeration gives palindromic and $γ$-positive polynomials. When the polynomials are $γ$-positive, we explain the $γ$-positivity through a group action analogous to Foata-Strehl valley hopping. Additionally, we study the avoidance of patterns in the relabelling orbit of $1213, 12112, 1231$ after a conjecture about their cardinalities by Laudone and give bijective proofs for the results.
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