Let $A_n$ be the coinvariant algebra of the symmetric group $S_n$. This algebra has vector space dimension $n!$. $A_n$ is the quotient algebra of the polynomial ring $K[x_1,...,x_n]$ by the elementary symmetric polynomials $e_i$. Since $e_1=x_1+....+x_n$, one can "eliminate" the variable $x_n$ to obtain an algebra with admissible relations.

Question: Is there a canonical monomial basis $b_p$ (in the literature) for $A_n$ parametrised by the elements of the symmetric group $p \in S_n$ such that the variable $x_n$ does not appear?

There is the Artin basis, see theorem 1.7 in https://www.jpswanson.org/talks/2017_coinvariant_zoo.pdf ,which can do this up to a choice of a bijection to the symmetric group elements and I wonder what bijection to the symmetric group elements one should use (probably there is a convention or a canonical way for this already).

One idea might be to "set" $r_i:=(i,i+1,...,n)$, then the symmetric group should be having also the elements $r_i$ as a basis meaning all elements of the symmetric group have the unique form $\Pi r_i^{a_i}$ as in the Artin basis. Is this the canonical way that should be used?