The Cardinal of Various Monoids of Transformations That Preserve a Uniform Partition

In this paper we give formulas for the number of elements of the monoids ORm×n of all full transformations on a finite chain with mn elements that preserve a uniform m-partition and preserve or reverse the orientation and for its submonoids ODm×n of all order-preserving or order-reversing elements, OPm×n of all orientation-preserving elements, Om×n of all order-preserving elements, O^+m×n of all extensive order-preserving elements and O^−m×n of all co-extensive order-preserving elements.