论文标题
COMTET统计数据的精制WILF等效度
Refined Wilf-equivalences by Comtet statistics
论文作者
论文摘要
我们通过统计量$ \ Mathsf {comp} $和$ \ Mathsf {iar} $启动了对精致的Wilf等效度的系统研究，其中$ \ Mathsf {comp}（π）$和$ \ Mathsf {IAR}（π）（π）（π）$是允许$ $ oftirt of Initial cuncon的$ uncon的数量和$ $ uncon的$ unconte $ unconter undere undere $ un。由于COMTET是第一个在他的书中考虑统计$ \ mathsf {comp} $的人，因此我们在此类课程中称为{\ em comtet统计}的任何$ \ mathsf {comp} $在一类排列上用$ \ mathsf {comp} $等分的统计量。这项工作是由Rubey在$ 321 $避免排列的三重等分结果的动机上，而第一和第三作者的最新结果是，$ \ mathsf {iar} $是可分离排列的comtet统计量。我们结果的一些亮点是： （1）双重comtet分布的对称性$（\ Mathsf {Comp}，\ Mathsf {iar}）$在几个加泰罗尼亚和Schröder类上，保留了左右最大值的值。 （2）$ \ mathsf {comp} $的完整分类和$ \ mathsf {iar} $ - wilf- equivalences for Length for Length $ 3 $模式和一对长度$ 3 $模式。计算$（\ Mathsf {des}，\ Mathsf {iar}，\ Mathsf {Comp}）$在这些模式上生成函数的这些模式避免了类和可分离排列。 （3）王统计$ \ mathsf {iar} $的进一步完善，是王最近的下降双重下降 - wilf等价之间可分离排列与$（2413,4213）$ - 避免排列的$（2413,4213）。
We launch a systematic study of the refined Wilf-equivalences by the statistics $\mathsf{comp}$ and $\mathsf{iar}$, where $\mathsf{comp}(π)$ and $\mathsf{iar}(π)$ are the number of components and the length of the initial ascending run of a permutation $π$, respectively. As Comtet was the first one to consider the statistic $\mathsf{comp}$ in his book {\em Analyse combinatoire}, any statistic equidistributed with $\mathsf{comp}$ over a class of permutations is called by us a {\em Comtet statistic} over such class. This work is motivated by a triple equidistribution result of Rubey on $321$-avoiding permutations, and a recent result of the first and third authors that $\mathsf{iar}$ is a Comtet statistic over separable permutations. Some highlights of our results are: (1) Bijective proofs of the symmetry of the double Comtet distribution $(\mathsf{comp},\mathsf{iar})$ over several Catalan and Schröder classes, preserving the values of the left-to-right maxima. (2) A complete classification of $\mathsf{comp}$- and $\mathsf{iar}$-Wilf-equivalences for length $3$ patterns and pairs of length $3$ patterns. Calculations of the $(\mathsf{des},\mathsf{iar},\mathsf{comp})$ generating functions over these pattern avoiding classes and separable permutations. (3) A further refinement by the Comtet statistic $\mathsf{iar}$, of Wang's recent descent-double descent-Wilf equivalence between separable permutations and $(2413,4213)$-avoiding permutations.
