Identities Derived from Noncrossing Partitions of Type B

TL;DR

This paper explores type B noncrossing partitions to derive new identities and involutions related to Narayana polynomials, extending known combinatorial results to a broader algebraic context.

Contribution

It introduces type B analogues of Coker's identities, involutions, and refinements, expanding the combinatorial understanding of Narayana polynomials.

Findings

Type B identities analogous to Coker's identities are established.

A parity reversing involution for type B Narayana numbers is constructed.

Type B analogues of Mansour and Sun's identities are provided.

Abstract

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.