AI 中文总结
该研究计数广义交替排列中避免长度为3的经典与vincular模式的元素个数，通过双射方法得到相关闭式结果，涉及完全k叉树森林、RSK等模型及多种模式类的计数。
AI 中文摘要
设 k 为不小于 2 的整数，D_{N,k} 为集合 {1,…,N} 中下降集恰好为 {k,2k,…,k*floor((N-1)/k)} 的排列构成的集合。我们计数 D_{N,k} 中避免每个长度为 3 的经典模式和每个 vincular 模式的元素个数。对于经典模式，我们给出从避免 132 模式和 231 模式的类到完全 k 叉树有序森林的递归双射，得到 Raney 数；避免 213 模式和 312 模式的类通过完成最后一个块并应用逆补对称性从这些森林双射得到；剩余的经典情况 321 用 RSK 表示。对于 vincular 模式，我们计数 6 个完全连续模式和 12 个恰好有一个邻接的模式，闭式结果伴随双射模型：Catalan、Raney、Fuss-Catalan 和 RSK 情况是经典双射的自然 k 叉或固定下降扩展，而乘积和偏序集情况由邻接条件强制的块插入和记录/树偏序集编码产生。
英文摘要
Let k be an integer at least 2, and let D_{N,k} be the set of permutations of {1,...,N} whose descent set is exactly {k, 2k, ..., k*floor((N-1)/k)}. We enumerate the elements of D_{N,k} avoiding each classical and each vincular pattern of length three. For classical patterns, we give recursive bijections from the 132- and 231-avoiding classes to ordered forests of complete k-ary trees, obtaining the Raney number. The 213- and 312-avoiding classes are obtained from these forest bijections by completing the last block and applying reverse-complement symmetry. The remaining classical case 321 is expressed by RSK. For vincular patterns, we enumerate the six fully consecutive patterns and the twelve patterns with exactly one adjacency. The closed-form results are accompanied by bijective models: the Catalan, Raney, Fuss-Catalan, and RSK cases are natural k-ary or fixed-descent extensions of classical bijections, while the product and poset cases arise from block-insertion and record/tree-poset encodings forced by the adjacency conditions.
Comments13 pages
