  In the context of statistical physics, Chandrasekharan and Wiese recently
introduced the \emph{fermionant} $\Ferm_k$, a determinant-like quantity where
each permutation $\pi$ is weighted by $-k$ raised to the number of cycles in
$\pi$. We show that computing $\Ferm_k$ is #P-hard under Turing reductions for
any constant $k > 2$, and is $\oplusP$-hard for $k=2$, even for the adjacency
matrices of planar graphs. As a consequence, unless the polynomial hierarchy
collapses, it is impossible to compute the immanant $\Imm_\lambda \,A$ as a
function of the Young diagram $\lambda$ in polynomial time, even if the width
of $\lambda$ is restricted to be at most 2. In particular, if $\Ferm_2$ is in
P, or if $\Imm_\lambda$ is in P for all $\lambda$ of width 2, then $\NP
\subseteq \RP$ and there are randomized polynomial-time algorithms for
NP-complete problems.
