On the Enumeration of Noncrossing Partitions with Fixed Points

Abstract

New results on the enumeration of noncrossing partitions with \(m\) fixed points are presented, using an enumeration polynomial \(P_m(x_1, x_2, \ldots, x_m)\). The double sequence of the coefficients \(a_{m,k}\) of each \(x^k_i\) in \(P_m\) is endowed with some important structural properties, which are used in order to determine the coefficient of each \(x^k_ix^l_j\) in \(P_m\).