LARGE SCHRÖDER PATHS BY TYPES AND SYMMETRIC FUNCTIONS

Title & Authors
LARGE SCHRÖDER PATHS BY TYPES AND SYMMETRIC FUNCTIONS
An, Su Hyung; Eu, Sen-Peng; Kim, Sangwook;

Abstract
In this paper we provide three results involving large Schr$\small{\ddot{o}}$der paths. First, we enumerate the number of large Schr$\small{\ddot{o}}$der 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 Schr$\small{\ddot{o}}$der paths. These results extend their counterparts of Kreweras and Armstrong-Eu on Dyck paths respectively.
Keywords
Schr$\small{\ddot{o}}$der paths;partial horizontal strips;sparse noncrossing partitions;elementary symmetric functions;
Language
English
Cited by
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