Proof of Dilks' bijectivity conjecture on Baxter permutations

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 numbers1[n+11]q[n+12]q∑k=0n−1q3(k+12)[n+1k]q[n+1k+1]q[n+1k+2]qtk, where [nk]q denote the q-binomial coefficients.