Classification of $\mathscr{L}$-cross-sections of the finite symmetric semigroup up to isomorphism

Let $\mathscr{T}_n$ be the symmetric semigroup of full transformations on a finite set with $n$ elements. In the paper we give a counting formula for the number of $\mathscr{L}$-cross-sections of $\mathscr{T}_n$ and classify all $\mathscr{L}$-cross-sections of $\mathscr{T}_n$ up to isomorphism.