### Electronic Research Archive

2021, Issue 5: 2877-2913. doi: 10.3934/era.2021018
Special Issues

# Refined Wilf-equivalences by Comtet statistics

• Received: 01 May 2020 Revised: 01 November 2020 Published: 15 March 2021
• Primary: 05A05, 05A15, 05A19; Secondary: 05C05

• 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 Analyse combinatoire, any statistic equidistributed with ${\mathsf{comp}}$ 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 ${\mathsf{iar}}$ is a Comtet statistic over separable permutations. Some highlights of our results are:

● Bijective proofs of the symmetry of the joint distribution $({\mathsf{comp}}, {\mathsf{iar}})$ over several Catalan and Schröder classes, preserving the values of the left-to-right maxima.

● 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.

● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and $(2413, 4213)$-avoiding permutations by the Comtet statistic ${\mathsf{iar}}$.

Citation: Shishuo Fu, Zhicong Lin, Yaling Wang. Refined Wilf-equivalences by Comtet statistics[J]. Electronic Research Archive, 2021, 29(5): 2877-2913. doi: 10.3934/era.2021018

### Related Papers:

• 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 Analyse combinatoire, any statistic equidistributed with ${\mathsf{comp}}$ 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 ${\mathsf{iar}}$ is a Comtet statistic over separable permutations. Some highlights of our results are:

● Bijective proofs of the symmetry of the joint distribution $({\mathsf{comp}}, {\mathsf{iar}})$ over several Catalan and Schröder classes, preserving the values of the left-to-right maxima.

● 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.

● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and $(2413, 4213)$-avoiding permutations by the Comtet statistic ${\mathsf{iar}}$.

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