Combinatorial results for order-preserving and order-decreasing transformations

Let O-n and C-n be the semigroup of all order-preserving transformations and of all order-preserving and order-decreasing transformations on the finite set X-n = {1,2, ..., n}, respectively. Let Fix(alpha) = {x is an element of X-n : x alpha = x} for any transformation alpha. In this paper; for any Y subset of X-n, we find the cardinalities of the sets O-n,O-Y = {alpha is an element of O-n : Fix(alpha) = Y} and C-n,C-Y = {alpha is an element of C-n : Fix(alpha) = Y}. Moreover; we find the numbers of transformations of O-n and C-n with r fixed points.