Refined Wilf-equivalences by Comtet statistics

We launch a systematic study of the refined Wilf-equivalences by the statistics \begin{document}$ {\mathsf{comp}} $\end{document} and \begin{document}$ {\mathsf{iar}} $\end{document} , where \begin{document}$ {\mathsf{comp}}(\pi) $\end{document} and \begin{document}$ {\mathsf{iar}}(\pi) $\end{document} are the number of components and the length of the initial ascending run of a permutation \begin{document}$ \pi $\end{document} , respectively. As Comtet was the first one to consider the statistic \begin{document}$ {\mathsf{comp}} $\end{document} in his book Analyse combinatoire, any statistic equidistributed with \begin{document}$ {\mathsf{comp}} $\end{document} over a class of permutations is called by us a Comtet statistic over such class. This work is motivated by a triple equidistribution result of Rubey on \begin{document}$ 321 $\end{document} -avoiding permutations, and a recent result of the first and third authors that \begin{document}$ {\mathsf{iar}} $\end{document} is a Comtet statistic over separable permutations. Some highlights of our results are: ● Bijective proofs of the symmetry of the joint distribution \begin{document}$ ({\mathsf{comp}}, {\mathsf{iar}}) $\end{document} over several Catalan and Schroder classes, preserving the values of the left-to-right maxima. ● A complete classification of \begin{document}$ {\mathsf{comp}} $\end{document} - and \begin{document}$ {\mathsf{iar}} $\end{document} -Wilf-equivalences for length \begin{document}$ 3 $\end{document} patterns and pairs of length \begin{document}$ 3 $\end{document} patterns. Calculations of the \begin{document}$ ({\mathsf{des}}, {\mathsf{iar}}, {\mathsf{comp}}) $\end{document} generating functions over these pattern avoiding classes and separable permutations. ● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and \begin{document}$ (2413, 4213) $\end{document} -avoiding permutations by the Comtet statistic \begin{document}$ {\mathsf{iar}} $\end{document} .