Two Statistics Linking Dyck Paths and Non-crossing Partitions

Haijian Zhao, Zheyuan Zhong
2011 Electronic Journal of Combinatorics  
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.
doi:10.37236/570 fatcat:uo3eqr76wjfw5joawiiounjknq