# Permanent

Permanent[m]

gives the permanent of the square matrix m.

# Details and Options • Permanent works with both numeric and symbolic matrices.
• The permanent of an matrix m is given by , where the are the permutations of elements.

# Examples

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## Basic Examples(1)

Find the permanent of symbolic matrices:

## Scope(1)

Use exact arithmetic to compute the permanent:

Use machine arithmetic:

Use 40-digit precision arithmetic:

## Applications(3)

The permanent of a square matrix of all ones is the factorial of the dimension:

The permanent of a square matrix of all ones minus the identity matrix counts the number of derangements of the corresponding dimension:

Given n sets, each containing a subset of (1 n), the number of ways to choose a distinct element from each subset is equal to the permanent of the 01 matrix where the (i,j) position contains a 1 exactly when subset i contains j:

There are two ways to create sets with distinct elements from each subset:

Confirm the result by explicitly constructing these sets:

## Properties & Relations(4)

The permanent is given by :

The permanent is a polynomial of its entries. Degree 2 for a matrix:

Degree 3 for a matrix etc.:

The determinant Det has the same terms as the permanent, except for sign changes:

The permanent is the outer product of the matrix rows, with terms having the repeated column index removed:

## Possible Issues(1)

Computing the permanent becomes slow even at modest dimension:

Introduced in 2015
(10.2)