Noncrossing Partitions with Fixed Points Having Specific Properties

Abstract

The noncrossing partitions with fixed points have been introduced and studied in the literature. In this paper, as their continuations, we derive expressions for \(f_m(x_1, 0^\mu, x_{\mu+2},0^\rho,x_{\mu+\mu+3},0^{m-\mu-\rho-3})\),and \(f_{m}(x_1,x_2, 0^\mu, x_{\mu+3},0^\rho,x_{\mu+\mu+3},0^{\rho+\mu+4},0^{m-\rho-\mu-4}\), are given,respectively. Moreover, we introduce noncrossing partitions with fixed points having specific property \(\mathcal{P}\) and describe their enumeration through a multivariable function \(f_m^\mathcal{P}(x_1, x_2, \ldots, x_m)\). Additionally, we obtain counting formulas for \(f_m^\mathcal{P}(x_1, 0^{m-1})\) and \(f_m^\mathcal{P}(x_1, x_2, 0^{m-2})\) for various properties \(\mathcal{P}\).