  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.
