ryser -- compute permanent using Ryser's formula

Synopsis

Usage:

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

Description

Uses Ryser's inclusion-exclusion formula (see page 27 of Ryser, Herbert John (1963), Combinatorial Mathematics, The Carus mathematical monographs, The Mathematical Association of America).

Let $M=(m_{i,j})$ be an nxn matrix. Then $perm(M)=\sum_{S\subseteq [n]} (-1)^{\mid S\mid} \prod_{i=1}^n \sum_{j\in S^c}m_{i,j}$.

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

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 : ryser 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 : ryser 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 : ryser M

o8 = 450

Caveat

Computationally intensive for moderate size matrices.

Ways to use ryser :

ryser(Matrix)

For the programmer