 Ryser's formula for the permanent

For an $n\times n$ matrix $A$, Ryser's formula for the permanent isExpanding a product chooses one column for each row. The alternating subset sum cancels each choice that fails to use every column, by inclusion-exclusion; the surviving choices are exactly the permutations in the permanent.

For a two-by-two matrix, the formula becomes $(a+b)(c+d)-ac-bd=ad+bc$. Maintaining row sums while enumerating subsets in Gray-code order gives an $O(n{2}^n)$ arithmetic algorithm, much smaller than enumerating $n!$ permutations, although still exponential in $n$.