Bernstein filtration ($F_j{A}_n=\mathrm{span}\{x^{\mathbf{a}}{\partial }^{\mathbf{b}}:|\mathbf{a}|+|\mathbf{b}|\le j\}$)

Give both position and differentiation generators of a Weyl algebra degree one. The ordered monomial basis gives $\dim F_j{A}_n=\left(\genfrac{}{}{0ex}{}{j+2n}{2n}\right)$ and ${\mathrm{gr}}_F{A}_n$ equal to a polynomial ring in $2n$ variables. Commuting a filtered element with any generator lowers its degree by at least one.