Theory of Computing ------------------- Title : The Complexity of the Fermionant and Immanants of Constant Width Authors : Stephan Mertens and Cristopher Moore Volume : 9 Number : 6 Pages : 273-282 URL : https://theoryofcomputing.org/articles/v009a006 Abstract -------- In the context of statistical physics, Chandrasekharan and Wiese recently introduced the _fermionant_ $Ferm_k$, a determinant-like function of a matrix 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 polynomial-time Turing reductions for any constant $k > 2$, and is $\oplusP$-hard for $k=2$, where both results hold even for the adjacency matrices of planar graphs. As a consequence, unless the polynomial-time 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, unless $NP \subseteq RP$, $Ferm_2$ is not in P, and there are Young diagrams $\lambda$ of width $2$ such that $\Imm_\lambda$ is not in P.