| Pattern avoidance is one of the important research topics of enumerative combinatorics,and its modern research originated from the work of Rotem,Rogers and Knuth.Baxter permutations are a kind of classical pattern avoiding permutations,which were first proposed by Glen Baxter when studying whether a pair of commutative continuous functions have common fixed points.The counting formula of Baxter permutations was first proved algebraically by Chung,Graham,Hogatt and Kleiman through introducing generating tree method.The first bijective proof of this counting formula was first given by Viennot,who used the classical Franccon-Viennot bijection to map Baxter permutations to non-intersecting triple of lattice paths.Based on three classical descent statistics of permutations,Dilks proposed a new conjectured bijection between Baxter permutations and non-intersecting triples of lattice paths in 2015,which gives more combinatorial meaning for Baxter permutations.The main contribution of this paper is to prove the bijectivity conjecture of Baxter permutations,which leads to several related counting results.In this paper,we first review three equivalent characterizations of Baxter permutations,and then introduce some known combinatorial bijections between Baxter permutations and twin binary trees,refined Young tableaus and non-intersecting triples of lattice paths.Finally,we give a detailed proof of the bijectivity conjecture of Baxter permutations proposed by Dilks.As a result,we obtain a permutation interpretation of the(q,t)analog of the Baxter numbers... |
