Noncrossing Combination With Fixed Points

In this paper, we study the enumeration of noncrossing partitions with fixed points. The expressions of f_m(x₁, x₂, x₃, 0, 0,…, 0) and f_m(x₁, x₂, 0,…, 0, x_ρ+3, 0,…, 0) are found and a new proof of the expression of f_m(x₁, x₂, 0, 0,…, 0) is obtained using diophantine equations. We also study the enumeration of noncrossing matchings with fixed points, and obtain some interesting results which are connected with Catalan numbers, e. g. g_n(1, 1, 1,…, 1)=C_n. We get the recursive formula of g_n(x₁, x₂,…, x_n) and find out the expression of g_n(x₁, x₂,…, x_k, 0, 0,…, 0) which can be expressed as 2^rC_i1 multiply from j=2 to t (C_ij+1).