Definition (Permanent of a matrix). Let $A$ be an $n\times n$ matrix over $ℝ$. The permanent of $A$, denoted $\mathrm{per}(A)$, is

$$\sum _{\sigma \in S_n}\prod _{i=1}^n{A}_{i\sigma (i)},$$

i.e. “the determinant without the signs”.