Refined Wilf-equivalences by Comtet statistics

Abstract

<p style='text-indent:20px;'>We launch a systematic study of the refined Wilf-equivalences by the statistics <inline-formula><tex-math id="M1">\begin{document}$ {\mathsf{comp}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ {\mathsf{iar}} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ {\mathsf{comp}}(\pi) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ {\mathsf{iar}}(\pi) $\end{document}</tex-math></inline-formula> are the number of components and the length of the initial ascending run of a permutation <inline-formula><tex-math id="M5">\begin{document}$ \pi $\end{document}</tex-math></inline-formula>, respectively. As Comtet was the first one to consider the statistic <inline-formula><tex-math id="M6">\begin{document}$ {\mathsf{comp}} $\end{document}</tex-math></inline-formula> in his book <i>Analyse combinatoire</i>, any statistic equidistributed with <inline-formula><tex-math id="M7">\begin{document}$ {\mathsf{comp}} $\end{document}</tex-math></inline-formula> over a class of permutations is called by us a <i>Comtet statistic</i> over such class. This work is motivated by a triple equidistribution result of Rubey on <inline-formula><tex-math id="M8">\begin{document}$ 321 $\end{document}</tex-math></inline-formula>-avoiding permutations, and a recent result of the first and third authors that <inline-formula><tex-math id="M9">\begin{document}$ {\mathsf{iar}} $\end{document}</tex-math></inline-formula> is a Comtet statistic over separable permutations. Some highlights of our results are:</p><p style='text-indent:20px;'>● Bijective proofs of the symmetry of the joint distribution <inline-formula><tex-math id="M10">\begin{document}$ ({\mathsf{comp}}, {\mathsf{iar}}) $\end{document}</tex-math></inline-formula> over several Catalan and Schröder classes, preserving the values of the left-to-right maxima.</p><p style='text-indent:20px;'>● A complete classification of <inline-formula><tex-math id="M11">\begin{document}$ {\mathsf{comp}} $\end{document}</tex-math></inline-formula>- and <inline-formula><tex-math id="M12">\begin{document}$ {\mathsf{iar}} $\end{document}</tex-math></inline-formula>-Wilf-equivalences for length <inline-formula><tex-math id="M13">\begin{document}$ 3 $\end{document}</tex-math></inline-formula> patterns and pairs of length <inline-formula><tex-math id="M14">\begin{document}$ 3 $\end{document}</tex-math></inline-formula> patterns. Calculations of the <inline-formula><tex-math id="M15">\begin{document}$ ({\mathsf{des}}, {\mathsf{iar}}, {\mathsf{comp}}) $\end{document}</tex-math></inline-formula> generating functions over these pattern avoiding classes and separable permutations.</p><p style='text-indent:20px;'>● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and <inline-formula><tex-math id="M16">\begin{document}$ (2413, 4213) $\end{document}</tex-math></inline-formula>-avoiding permutations by the Comtet statistic <inline-formula><tex-math id="M17">\begin{document}$ {\mathsf{iar}} $\end{document}</tex-math></inline-formula>.</p>