Potent elements in semigroups of transformations

Let Pn and Tn denote the semigroup of all partial transformations and the semigroup of all transformations on the set Xn ={1,2,?,n}, respectively. Let On and Dn denote the semigroup of all order-preserving elements of Tn and the semigroup of all order-decreasing elements of Tn, respectively. Let Cn = On ? Dn.We give formulas for the number of m-potent elements, the number of (m, r)-potent elements and the number of m-nilpotent elements of the semigroup Pn. We show that the number of elements of On which has exactly m fixed points is the coefficient of xn-m in f 2m(x) where f (x) is the generating function of the Catalan numbers. Moreover, we compute the coefficient of xn in f m(x) for given n,m ? N.Let us define the setsCn,r = {? ? On : ? is s-potent where s ? r and Fix (?) = {1}},Wn,r = {? ? On : ? is s-potent where s ? r and |Fix (?)| = 1},Un,r = {? ? On : ? is s-potent where s ? r}.Cn,r forms a subsemigroup of the semigroup On. Let us denote the generating sets of the numbers |Cn,r|, |Wn,r|, |Un,r| by Cr (x), Wr (x), Ur (x), respectively. Also we denote the set of ordered trees with depth at most r and n number of nodes by ?n,r. We establish some relations among the functions Cr (x), Wr (x), Ur (x) and thereby among the numbers |Cn,r|, |Wn,r|, |Un,r|. We establish a bijective correspondence between Cn,r and ?n,r. Besides, we obtain formulas for Cr (x), Wr (x), Ur (x).