Proof of Dilks' bijectivity conjecture on Baxter permutations

TL;DR

This paper proves Dilks' conjecture that establishes a bijection between Baxter permutations and certain lattice paths, connecting it with the Fran ext{c}on--Viennot bijection and providing a permutation interpretation of a $(t,q)$-analog of Baxter numbers.

Contribution

We prove Dilks' bijectivity conjecture for Baxter permutations and connect it with the Fran ext{c}on--Viennot bijection, offering a new permutation interpretation of Baxter numbers.

Findings

Confirmed Dilks' bijection conjecture for Baxter permutations.

Connected the bijection with the Fran ext{c}on--Viennot bijection.

Provided a permutation interpretation of the $(t,q)$-analog of Baxter numbers.

Abstract

Baxter permutations originally arose in studying common fixed points of two commuting continuous functions. In 2015, Dilks proposed a conjectured bijection between Baxter permutations and non-intersecting triples of lattice paths in terms of inverse descent bottoms, descent positions and inverse descent tops. We prove this bijectivity conjecture by investigating its connection with the Fran\c{c}on--Viennot bijection. As a result, we obtain a permutation interpretation of the $(t,q)$-analog of the Baxter numbers $$ \frac{1}{{n+1\brack 1}_q{n+1\brack 2}_q}\sum_{k=0}^{n-1}q^{3{k+1\choose2}}{n+1\brack k}_q{n+1\brack k+1}_q{n+1\brack k+2}_qt^k, $$ where ${n\brack k}_q$ denote the $q$-binomial coefficients.