Identities derived from noncrossing partitions of type \(B\) (Q547789)

Summary: Based on weighted noncrossing partitions of type \(B\), we obtain type \(B\) analogues of Coker's identities on the Narayana polynomials. A parity reversing involution is given for the alternating sum of Narayana numbers of type \(B\). Moreover, we find type \(B\) analogues of the refinements of Coker's identities due to Chen, Deutsch and Elizalde. By combinatorial constructions, we provide type \(B\) analogues of three identities of Mansour and Sun also on the Narayana polynomials.