We launch a systematic study of the refined Wilf-equivalences by the statistics $mathsf{comp}$ and $mathsf{iar}$, where $mathsf{comp}(pi)$ and $mathsf{iar}(pi)$ are the number of components and the length of the initial ascending run of a permutation $pi$, 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 Schroder 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 Wangs recent descent-double descent-Wilf equivalence between separable permutations and $(2413,4213)$-avoiding permutations.
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