Families of parking functions counted by the schröder and baxter numbers

We define two new families of parking functions: one counted by Schröder numbers and the other by Baxter numbers. These families both include the well-known class of non-decreasing parking functions, which is counted by Catalan numbers and easily represented by Dyck paths, and they both are included in the class of underdiagonal sequences, which are bijective to permutations. We investigate their combinatorial properties exhibiting bijections between these two families and classes of lattice paths (Schröder paths and triples of non-intersecting lattice paths) and discovering a link between them and some classes of pattern avoiding permutations. Then, we provide a quite natural generalization for each of these families that results in some enumeration problems tackled by ECO method.