Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain

Let ${\cal T}_n$ be the full symmetric semigroup on $X_n = \{1, 2,..., n\}$ and let ${\cal OCT}_n$ and ${\cal ORCT}_n$ be its subsemigroups of order-preserving and order-preserving or order-reversing full contraction mappings of $X_n$, respectively. In this paper we investigate the cardinalities of some equivalences on ${\cal OCT}_n$ and ${\cal ORCT}_n$ which lead naturally to obtaining the orders of these subsemigroups.