A note on symmetric orderings

Let ${\widehat{A}}_n$ be the completion by the degree of a differential operator of the $n$-th Weyl algebra with generators $x_1,\dots ,x_n,{\mathrm{\partial }}^1,\dots ,{\mathrm{\partial }}^n$. Consider $n$ elements $X_1,\dots ,X_n$ in ${\widehat{A}}_n$ of the form $$X_i=x_i+\sum _{K=1}^{\mathrm{\infty }}\sum _{l=1}^n\sum _{j=1}^n{x}_l{p}_{ij}^{K-1,l}(\mathrm{\partial }){\mathrm{\partial }}^j,$$ where $p_{ij}^{K-1,l}(\mathrm{\partial })$ is a degree $(K-1)$ homogeneous polynomial in ${\mathrm{\partial }}^1,\dots ,{\mathrm{\partial }}^n$, antisymmetric in subscripts $i,j$. Then for any natural $k$ and any function $i : {1,\ldots,k}\to{1,\ldots,n}$weprove$$ \sum_{\sigma \in \Sigma(k)} X_{i_{\sigma(1)}}\cdots X_{i_{\sigma(k)}}\triangleright 1 = k! ,x_{i_1}\cdots x_{i_k}, $$where$\Sigma(k)$isthesymmetricgroupon$k$lettersand$\triangleright$ denotes the Fock action of the ${\widehat{A}}_n$ on the space of (commutative) polynomials.