assume the result for $r\leq m-1$. Since the permutation group on $m$ letters is generated by the transpositions $(i,i+1)$ for $i=1,\dots,m-1$, we may assume without loss of generality that $$\tilde{\mu} = ~:a_1 \cdots a_{i-1} a_{i+1} a_i a_{i+2} \cdots a_m:.$$ If $i>1$, we have $\mu - \tilde{\mu} = ~:a_1 \cdots a_{i-1} f:$, where $f =~ :a_i \cdots a_m:~ -~: a_{i+1}a_i a_{i+2} \cdots a_m$, which lies in $(\cM_{-n})_{(2m-2i+2)}$. Since each term of $f$ has degree at least 2, it follows that $\mu - \tilde{\mu}$ can be expressed in the form $\sum_{k=i}^{m-1}E^{2k}$. Since $i>1$, there is no term of degree $2$. Given any rearrangement $\mu - \tilde{\mu}=\sum_{k=1}^{m-1}F^{2k}$, it follows from our inductive hypothesis that the term $F^2$ is a second derivative.

Suppose next that $i=1$, so that $\tilde{\mu} = ~:a_2 a_1 a_3 \cdots a_m:$. Define $$\nu = \ :(:a_1 a_2:) a_3 \cdots a_m:,\ \ \ \ \ \ \tilde{\nu} = \ :(:a_2 a_1:) a_3 \cdots a_m:,$$ and note that $\nu  - \tilde{\nu} = \ : (:a_1 a_2: -: a_2 a_1: )f:$, where $f = \ :a_3 \cdots a_m:$. By (\[reari\]), $:a_1 a_2: - : a_2 a_1:$ is homogeneous of degree 2, so $\nu  - \tilde{\nu}$ is a linear combination of monomials of degree 