Combinatorics of a generalized Narayana identity

The Narayana identity is a well-known formula that expresses the classical Catalan numbers as sums of the ordinary Narayana numbers. In this paper we generalize the Narayana identity to a family of Riordan arrays including the array of ballot numbers, the classical Catalan triangle and several generalized Catalan triangles recently studied. A combinatorial description based on non-crossing partitions is given for this identity, for the column-recursive rule, and for the Sheffer sequence associated with any array of the family.