Consider the ring of polynomials $R:=\mathbb{Z}[x_1,x_2,x_3]$. Define the operators $E, I:R\rightarrow R$ by $Ef(x_1,x_2,x_3)=f(x_1-1,x_2,x_3)$ and the identity $If=f$.

Let $\mathcal{L}:R\rightarrow R$ be the operator given by $$\mathcal{L}f=[(x_1+x_2)(x_1+x_3)E-x_1^2I]f.$$

Let $1$ stand for the constant function $f(x_1,x_2,x_3)=1$ and $\mathcal{L}^2f=\mathcal{L}(\mathcal{L}f)$, etc.

CLAIM.Experiments suggest that $\mathcal{L}^n1$ is always a symmetric polynomial in $R$. Any proof?

**EDIT.** This has found a resolution (see Pietro Majer's answer).

For example, $\mathcal{L} 1=e_2$ and $\mathcal{L}^21=e_2^2-e_1e_2+e_3$ where $e_1=x_1+x_2+x_3, e_2=x_1x_2+x_1x_3+x_2x_3, e_3=x_1x_2x_3$ are the standard elementary symmetric polynomials.

QUESTIONS.(EDIT) These did not find a definitive answer (apart from Brendan McKay's evidence and argument).(1) Are there other orbits of symmetric polynomials under $\mathcal{L}$?

(2) Are there other non-trivial operators with similar property over $R$?

(3) What about over rings of many more variables?