= Forgotten symmetric function
{title2=$f_\lambda=(-1)^{|\lambda|-\ell(\lambda)}\omega(m_\lambda)$}
The coefficient of $m_\mu$ in $f_\lambda$ counts distinct rearrangements of the parts of $\lambda$ whose prefix-sum set contains the prefix sums of $\mu$. Indeed, <Hall inner product of symmetric functions> duality between $m_\lambda,h_\lambda$ turns this coefficient into $(-1)^{|\lambda|-\ell(\lambda)}$ times the coefficient of $h_\lambda$ in $e_\mu$. Expanding $E(t)=H(-t)^{-1}$ expresses each $e_{\mu_j}$ as a signed sum of compositions of $\mu_j$. Concatenating these compositions gives precisely the counted rearrangements, and every sign is $(-1)^{|\lambda|-\ell(\lambda)}$.
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