Large Schr\"oder paths by types and symmetric functions
Bull. Korean Math. Soc. 2014 Vol. 51, No. 4, 1229-1240
Printed July 1, 2014
Su Hyung An, Sen-Peng Eu, and Sangwook Kim
Yonsei University, National Taiwan Normal University, Chonnam National University
Abstract : In this paper we provide three results involving large Schr\"oder paths. First, we enumerate the number of large Schr\"oder 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\"oder paths. These results extend their counterparts of Kreweras and Armstrong-Eu on Dyck paths respectively.
Keywords : Schr\"oder paths, partial horizontal strips, sparse noncrossing partitions, elementary symmetric functions
MSC numbers : Primary 05A15, 05E05
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