In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function of a matrix called the immanant.

The permanent of an n×n matrix A = (a_i,j) is defined as

$${\displaystyle \operatorname {perm} (A)=\sum _{\sigma \in S_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}.}$$

The sum here extends over all elements σ of the symmetric group S_n; i.e. over all permutations of the numbers 1, 2, ..., n.

For example,

$${\displaystyle \operatorname {perm} {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=ad+bc,}$$

and

$${\displaystyle \operatorname {perm} {\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}=aei+bfg+cdh+ceg+bdi+afh.}$$