A bijection for length-$5$ patterns in permutations

TL;DR

This paper constructs a bijection between two classes of pattern-avoiding permutations that preserves key statistics, leading to the proof of an enumerative conjecture and showing the algebraic nature of their generating function.

Contribution

It introduces a new bijection between specific pattern-avoiding permutations and proves an enumerative conjecture, also establishing the algebraic form of their generating function.

Findings

Bijection preserves five classical set-valued statistics.

Proof of an enumerative conjecture by Gao and Kitaev.

Generating function for the counting sequence is algebraic.

Abstract

A bijection between $(31245,32145,31254,32154)$-avoiding permutations and $(31425,32415,31524,32514)$-avoiding permutations is constructed, which preserves five classical set-valued statistics. Combining with two codings of permutations due respectively to Baril--Vajnovszki and Martinez--Savage proves an enumerative conjecture posed by Gao and Kitaev. Moreover, the generating function for the common counting sequence is proved to be algebraic.