Combinatorial results for certain semigroups of transformations preserving orientation and a uniform partition

Let TX be the full transformation semigroup on a set X and E be a non-trivial equivalence on X. The set TE(X)={f?TX:?(x,y)?E,(f(x),f(y))?E} is a subsemigroup of TX. For a finite totally ordered set X and a convex equivalence E on X, the set of all the orientation-preserving transformations in TE(X) forms a subsemigroup of TE(X) denoted by OPE(X). In this paper, under the hypothesis that the totally ordered set X is of cardinality mn(m,n?2) and the equivalence E has m classes such that each E-class contains n consecutive points, we calculate the cardinality of the semigroup OPE(X), and that of its idempotents.