The Permanent(A) function computes the permanent of A.

Similar to the Matrix determinant, the permanent P(A) of an n x n Matrix A can be defined in terms of a sum along any row or column, with unsigned minor expansion, by the following definition.

For any i in 1 .. n,

$P\left(A\right)=\mathrm{add}\left(A_{i,j}P\left(\mathrm{A\_}\left(i\,j\right)\right)\,j=1..n\right)$

where

A_(i, j) is the i, jth minor of A given by

$\mathrm{A\_}\left(i\,j\right)=A_{1..i-1,i+1..n,1..j-1,j+1..n}$

which is A with the ith row and jth column removed.

This definition differs from that of the Determinant only by the absence of alternating signs of the terms in the sum.

This function is part of the LinearAlgebra package, and so it can be used in the form Permanent(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[Permanent](..).

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$

$\mathrm{Permanent}\left(⟨⟨5\,-1⟩|⟨-2\,4⟩⟩\right)$

$22$

$A≔⟨⟨i\,l\,o⟩\left|⟨j\,m\,p⟩\right|⟨k\,n\,q⟩⟩$

$A≔\left[\begin{array}{ccc}i& j& k\\ l& m& n\\ o& p& q\end{array}\right]$

$\mathrm{Permanent}\left(A\right)$

$imq+inp+jlq+jno+klp+kmo$

$\mathrm{Determinant}\left(A\right)$

$imq-inp-jlq+jno+klp-kmo$