A group action on noncrossing partitions

Let NC n denote the set of noncrossing partitions of [ n ] = { 1 , 2 , … , n } . By defining a group action on the set NC n , we give a combinatorial interpretation of the expansion of the n -th Narayana polynomial on the basis { q k ( 1 + q ) n − 1 − 2 k } , k = 0 , 1 , … , ⌊ ( n − 1 ) / 2 ⌋ .