Given an integer $n\geq 1$ and an irreducible character $��_��$ of $S_{n}$ for some partition $��$ of $n$, the immanant $\mathrm{imm}_��:\mathbb{C}^{n\times n}\to\mathbb{C}$ maps matrices $A\in\mathbb{C}^{n\times n}$ to $\mathrm{imm}_��(A)=\sum_{��\in S_{n}}��_��(��)\prod_{i=1}^{n}A_{i,��(i)}$. Important special cases include the determinant and permanent, which are the immanants associated with the sign and trivial character, respectively. It is known that immanants can be evaluated in polynomial time for characters that are close to the sign character: Given a partition $��$ of $n$ with $s$ parts, let $b(��):=n-s$ count the boxes to the right of the first column in the Young diagram of $��$. For a family of partitions $��$, let $b(��):=\max_{��\in��}b(��)$ and write Imm$(��)$ for the problem of evaluating $\mathrm{imm}_��(A)$ on input $A$ and $��\in��$. If $b(��)

28 pages, to appear at STOC'21