2020/09/09 by Shishuo Fu, Fu, Shishuo, Zhicong Lin +3 · 2 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algorithms and Data Compression #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2009.04269
openalex publication_date 2020/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We launch a systematic study of the refined Wilf-equivalences by the statistics comp and iar, where comp(π) and 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 comp in his book \em Analyse combinatoire, any statistic equidistributed with 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 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 (comp,iar) over several Catalan and Schröder classes, preserving the values of the left-to-right maxima. (2) A complete classification of comp- and iar-Wilf-equivalences for length 3 patterns and pairs of length 3 patterns. Calculations of the (des,iar,comp) generating functions over these pattern avoiding classes and separable permutations. (3) A further refinement by the Comtet statistic iar, of Wang's recent descent-double descent-Wilf equivalence between separable permutations and (2413,4213)-avoiding permutations.