  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 Schr\"oder 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.
