Large schröder paths by types and symmetric functions

In this paper we provide three results involving large Schröder paths. First, we enumerate the number of large Schröder paths by type. Second, we prove that these numbers are the coefficients of a certain symmetric function defined on the staircase skew shape when expanded in elementary symmetric functions. Finally we define a symmetric function on a Fuss path associated with its low valleys and prove that when expanded in elementary symmetric functions the indices are running over the types of all Schröder paths. These results extend their counterparts of Kreweras and Armstrong-Eu on Dyck paths respectively.