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
open allclose allScope (2)
Options (2)
Method (1)
Modulus (1)
Compute a permanent using arithmetic modulo 47:
This is faster than computing Mod[Permanent[m],47]:
Applications (5)
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 0‐1 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:
Properties & Relations (5)
The permanent is a polynomial of its entries. Degree 2 for a matrix:
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:
The permanent is invariant under a symmetric permutation of rows and columns:
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