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http://dx.doi.org/10.4134/BKMS.2014.51.4.1229

LARGE SCHRÖDER PATHS BY TYPES AND SYMMETRIC FUNCTIONS  

An, Su Hyung (Department of Mathematics Yonsei University)
Eu, Sen-Peng (Department of Mathematics National Taiwan Normal University)
Kim, Sangwook (Department of Mathematics Chonnam National University)
Publication Information
Bulletin of the Korean Mathematical Society / v.51, no.4, 2014 , pp. 1229-1240 More about this Journal
Abstract
In this paper we provide three results involving large Schr$\ddot{o}$der paths. First, we enumerate the number of large Schr$\ddot{o}$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$\ddot{o}$der paths. These results extend their counterparts of Kreweras and Armstrong-Eu on Dyck paths respectively.
Keywords
Schr$\ddot{o}$der paths; partial horizontal strips; sparse noncrossing partitions; elementary symmetric functions;
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