# Refined Wilf-equivalences by Comtet statistics

We launch a systematic study of the refined Wilf-equivalences by the statistics πΌπππ and ππΊπ, where πΌπππ(Ο) and ππΊπ(Ο) are the number of components and the length of the initial ascending run of a permutation Ο, respectively. As Comtet was the first one to consider the statistic πΌπππ in his book Analyse combinatoire, any statistic equidistributed with πΌπππ over a class of permutations is called by us a 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 ππΊπ is a Comtet statistic over separable permutations. Some highlights of our results are: (1) Bijective proofs of the symmetry of the double Comtet distribution (πΌπππ,ππΊπ) over several Catalan and SchrΓΆder classes, preserving the values of the left-to-right maxima. (2) A complete classification of πΌπππ- and ππΊπ-Wilf-equivalences for length 3 patterns and pairs of length 3 patterns. Calculations of the (π½πΎπ,ππΊπ,πΌπππ) generating functions over these pattern avoiding classes and separable permutations. (3) A further refinement by the Comtet statistic ππΊπ, of Wang's recent descent-double descent-Wilf equivalence between separable permutations and (2413,4213)-avoiding permutations.

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