Forgotten symmetric function ($f_{\lambda }=(-1{)}^{|\lambda |-\ell (\lambda )}\omega (m_{\lambda })$)

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 $(-1{)}^{|\lambda |-\ell (\lambda )}$.