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.
Examplesopen allclose all
Compute a permanent using arithmetic modulo 47:
This is faster than computing Mod[Permanent[m],47]:
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:
Wolfram Research (2015), Permanent, Wolfram Language function, https://reference.wolfram.com/language/ref/Permanent.html (updated 2022).
Wolfram Language. 2015. "Permanent." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Permanent.html.
Wolfram Language. (2015). Permanent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Permanent.html