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.



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> 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?
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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?



Basis parametrized by the symmetric group elements for the coinvariant algebra
So the Artin basis already does this. There is a natural bijection of the symmetric group $S\_n$ with sequences $(a\_1,\ldots,a\_{n-1})$ of integers $a\_i$ with $0\leq a\_i\leq n-i$. Namely, let $w:[n]\to [n]$ be a permutation. Then define
$$a\_i(w)=\#\{j>i\mid w(i)>w(j)\}$$
This is known as the Lehmer code. It's a little bit more work to get back to the permutation from the Lehmer code, but it's still not very hard. For an algorithm, start with a $0$-indexed sequence $A^0=(1,2,\ldots,n)$. To construct $w$ given $(a\_1,\ldots,a\_{n-1})$, at iteration $i$ set $w(i)=A^{i-1}\_{a\_i}$ and ``pop'' the element at index $a\_i$ to obtain a new $0$-indexed sequence $A^i$. For the last index $w(n)$, the value will be the only remaining element in $A^{n-1}$.


