Find the permanent of a symbolic matrix:

The permanent of an exact matrix:

Find the permanent of a symbolic matrix:

Use exact arithmetic to compute the permanent:

Use machine arithmetic:

Use 40-digit precision arithmetic:

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:

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:

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:

Computing the permanent becomes slow even at modest dimension: