Permanent

In linear algebra, the permanent of an n-by-n matrix A=(a_i,j) is defined as

<math>\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}</math>

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

For example,

<math>\operatorname{per}\begin{pmatrix}a&b\\

c&d\end{pmatrix}=ad+bc.</math>

The definition of the permanent of A differs from that of the determinant of A in that the signatures of the permutations are not taken into account. If one views the permanent as a map that takes n vectors as arguments, then it is a multilinear map and it is symmetric (meaning that any order of the vectors results in the same permanent). A formula similar to Laplace's for the development of a determinant along a row or column is also valid for the permanent; all signs have to be ignored for the permanent.

Unlike the determinant, the permanent has no easy geometrical interpretation; it is mainly used in combinatorics. The permanent is not multiplicative. It is also not possible to use Gaussian elimination to compute the permanent; no fast algorithms for its computation are known.