Refined Wilf-equivalences by Comtet statistics

09/09/2020

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by   Shishuo Fu, et al.

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We launch a systematic study of the refined Wilf-equivalences by the statistics 𝖼𝗈𝗆𝗉 and 𝗂𝖺𝗋, where 𝖼𝗈𝗆𝗉(π) and 𝗂𝖺𝗋(π) are the number of components and the length of the initial ascending run of a permutation π, respectively. As Comtet was the first one to consider the statistic 𝖼𝗈𝗆𝗉 in his book Analyse combinatoire, any statistic equidistributed with 𝖼𝗈𝗆𝗉 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 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 (𝖼𝗈𝗆𝗉,𝗂𝖺𝗋) over several Catalan and Schröder classes, preserving the values of the left-to-right maxima. (2) A complete classification of 𝖼𝗈𝗆𝗉- and 𝗂𝖺𝗋-Wilf-equivalences for length 3 patterns and pairs of length 3 patterns. Calculations of the (𝖽𝖾𝗌,𝗂𝖺𝗋,𝖼𝗈𝗆𝗉) generating functions over these pattern avoiding classes and separable permutations. (3) A further refinement by the Comtet statistic 𝗂𝖺𝗋, of Wang's recent descent-double descent-Wilf equivalence between separable permutations and (2413,4213)-avoiding permutations.