glynn -- compute permanent using Glynn's formula

Synopsis

Usage:

F = glynn M
Inputs:
- M, a matrix, a square matrix in any commutative ring
Outputs:
- F, a ring element, the permanent of M

Description

Uses Glynn's formula (see Glynn, David G. (2010), "The permanent of a square matrix", European Journal of Combinatorics 31 (7): 1887–1891, doi:10.1016/j.ejc.2010.01.010).

Let $M=(m_{i,j})$ be an nxn matrix. Then $perm(M)=\sum_{\delta}\prod_{k=1}^n(\delta_k) \prod_{i=1}^n\sum_{j=1}^n \delta_j m_{i,j}$ where the outer summation is over $\delta\in \{ -1,+1\}^n$ with $\delta_1=1$.

For example, for the 3x3 generic matrix Glynn’s formula gives $2^2 perm(M) =(a + b + c)(d + e + f)(g + h + i) − (a − b + c)(d − e + f)(g − h + i) −(a + b − c)(d + e − f)(g + h − i) + (a − b − c)(d − e − f)(g − h − i)$.

Here is the permanent of a 3x3 generic matrix of variables.

i1 : R = QQ[vars(0..8)]

o1 = R

o1 : PolynomialRing

i2 : M = genericMatrix(R,a,3,3)

o2 = | a d g |
     | b e h |
     | c f i |

             3      3
o2 : Matrix R  <-- R

i3 : glynn M

o3 = c*e*g + b*f*g + c*d*h + a*f*h + b*d*i + a*e*i

o3 : R

Here is the permanent of a 4x4 generic matrix of variables.

i4 : R = QQ[vars(0..15)]

o4 = R

o4 : PolynomialRing

i5 : M = genericMatrix(R,a,4,4)

o5 = | a e i m |
     | b f j n |
     | c g k o |
     | d h l p |

             4      4
o5 : Matrix R  <-- R

i6 : glynn M

o6 = d*g*j*m + c*h*j*m + d*f*k*m + b*h*k*m + c*f*l*m + b*g*l*m + d*g*i*n +
     ------------------------------------------------------------------------
     c*h*i*n + d*e*k*n + a*h*k*n + c*e*l*n + a*g*l*n + d*f*i*o + b*h*i*o +
     ------------------------------------------------------------------------
     d*e*j*o + a*h*j*o + b*e*l*o + a*f*l*o + c*f*i*p + b*g*i*p + c*e*j*p +
     ------------------------------------------------------------------------
     a*g*j*p + b*e*k*p + a*f*k*p

o6 : R

Here is the permanent of a 3x3 matrix of integers.

i7 : M=matrix{{1,2,3},{4,5,6},{7,8,9}}

o7 = | 1 2 3 |
     | 4 5 6 |
     | 7 8 9 |

              3       3
o7 : Matrix ZZ  <-- ZZ

i8 : glynn M

o8 = 450

Caveat

Works in characteristic not equal to 2 because need to divide by $2^{n-1}$. Computationally intensive for moderate size matrices.

Ways to use glynn :

glynn(Matrix)

For the programmer