Combinatorial results for order-preserving and order-decreasing transformations

Let On and Cn be the semigroup of all order-preserving transformations and of all order-preserving and order-decreasing transformations on the finite set Xn = {1,2,...,n}, respectively. Let Fix(α)= {x element of Xn : xα = x} for any transformation α. In this paper, for any Y subseteq Xn , we find the cardinalities of the sets On, Y = {α element of On :Fix(α)= Y} and Cn,Y = {α element of Cn :Fix(α)= Y}. Moreover, we find the numbers of transformations of On and Cn with r fixed points.