Gr\"obner-Shirshov basis for the finitely presented algebras defined by permutation relations of symmetric type

In this paper, we give a Gr\"obner-Shirshov basis for the finitely presented semigroup algebra $\mathbf{k}[S_n(Sym_n)]$ defined by permutation relations of symmetric type. As an application, by the Composition-Diamond Lemma, we obtain normal forms of elements of momoid $S_n(Sym_n)$, which gives an answer to an open problem posted by F. Ced\'o, E. Jespers and J. Okni\'nski [7] for the symmetric group case.