Two Statistics Linking Dyck Paths and Non-crossing Partitions

Haijian Zhao, Zheyuan Zhong


We introduce a pair of statistics, maj and sh, on Dyck paths and show that they are equidistributed. Then we prove that this maj is equivalent to the statistics $ls$ and $rb$ on non-crossing partitions. Based on non-crossing partitions, we give the most obvious $q$-analogue of the Narayana numbers and the Catalan numbers.

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