A Note on Enumeration of Noncrossing Partitions

Abstract

In this paper, we study the enumeration of noncrossing partitions with fixed points. The expressions of \({f_m}(x_1, x_2,x_3, 0, \ldots, 0)\) and \({f_m}(x_1, x_2, 0, \ldots, 0, x_{p+3}, 0, \ldots, 0)\) are found, and a new proof of the expression of \({f_m}(x_1, x_2,0, 0, \ldots, 0)\) is obtained using diophantine equations.