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

Let ${\mathcal{T}}_n$ be the full symmetric semigroup on $X_n=\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ and let ${\mathcal{O}\mathcal{C}\mathcal{T}}_n$ and ${\mathcal{O}\mathcal{R}\mathcal{C}\mathcal{T}}_n$ 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 ${\mathcal{O}\mathcal{C}\mathcal{T}}_n$ and ${\mathcal{O}\mathcal{R}\mathcal{C}\mathcal{T}}_n$ which lead naturally to obtaining the orders of these subsemigroups.