Proof of Dilks' bijectivity conjecture on Baxter permutations

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 Françon--Viennot bijection. As a result, we obtain a permutation interpretation of the $(t,q)$-analog of the Baxter numbers $$\frac{1}{{\left[\genfrac{}{}{0ex}{}{n+1}{1}\right]}_q{\left[\genfrac{}{}{0ex}{}{n+1}{2}\right]}_q}\sum _{k=0}^{n-1}{q}^{3\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)}{\left[\genfrac{}{}{0ex}{}{n+1}{k}\right]}_q{\left[\genfrac{}{}{0ex}{}{n+1}{k+1}\right]}_q{\left[\genfrac{}{}{0ex}{}{n+1}{k+2}\right]}_q{t}^k,$$ where ${\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q$ denote the $q$-binomial coefficients.

arXiv comment11 pages, 3 figures