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.
  • Permanent[m,Modulus->n] computes the permanent modulo n.
  • Permanent supports a Method option. Possible settings include "MemoizedExpansion", "MixedCoefficient", "Glynn" and "Ryser". The default setting of Automatic switches among these methods depending on the matrix given.

Examples

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Basic Examples  (2)

Find the permanent of a symbolic matrix:

The permanent of an exact matrix:

Scope  (2)

Find the permanent of a symbolic matrix:

Use exact arithmetic to compute the permanent:

Use machine arithmetic:

Use 40-digit precision arithmetic:

Options  (2)

Method  (1)

Use the default method for numerical evaluation:

Obtain the result from all available methods:

Compare the timings of the four methods:

Modulus  (1)

Compute a permanent using arithmetic modulo 47:

This is faster than computing Mod[Permanent[m],47]:

Applications  (6)

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:

Use the permanent to compute the number of ways n queens can be placed on an n×n chessboard such that no two queens attack each other:

The permanental polynomial of a graph is defined as the permanent of id x-m, where m is the corresponding adjacency matrix and id is the identity matrix of appropriate size:

The permanental polynomial of a path graph with n vertices is equivalent to the Fibonacci polynomial :

The permanental polynomial of a cycle graph can also be expressed in terms of the Fibonacci polynomial:

Define a function for computing a Fourier matrix with unit normalization:

For even dimensions, the permanent is zero:

For odd dimensions, the permanent is always an integer:

For an odd prime p>3, the permanent of the p×p matrix is congruent to p!, modulo p3:

Properties & Relations  (10)

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 can be computed recursively via cofactor expansion along any row:

Or any column:

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

A matrix and its transpose have the same permanent:

The permanent is invariant under an arbitrary permutation of rows and columns:

The permanent of a matrix multiplied by a diagonal matrix is equal to the product of the permanent of the original matrix and the diagonal elements of the diagonal matrix:

The permanent of a triangular matrix is the product of its diagonal elements:

The permanent of a block diagonal matrix is the product of the permanents of the diagonal blocks:

Possible Issues  (1)

Computing the permanent becomes slow even at modest dimension:

Wolfram Research (2015), Permanent, Wolfram Language function, https://reference.wolfram.com/language/ref/Permanent.html (updated 2022).

Text

Wolfram Research (2015), Permanent, Wolfram Language function, https://reference.wolfram.com/language/ref/Permanent.html (updated 2022).

CMS

Wolfram Language. 2015. "Permanent." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Permanent.html.

APA

Wolfram Language. (2015). Permanent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Permanent.html

BibTeX

@misc{reference.wolfram_2024_permanent, author="Wolfram Research", title="{Permanent}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Permanent.html}", note=[Accessed: 17-July-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_permanent, organization={Wolfram Research}, title={Permanent}, year={2022}, url={https://reference.wolfram.com/language/ref/Permanent.html}, note=[Accessed: 17-July-2024 ]}