Handbook of Linear Algebra, 2nd Edition

Chapter 43

Permanents

Ian M. Wanless

Monash University

The permanent is a matrix function introduced (independently) by Cauchy and Binet in 1812. At first sight it seems to be a simplified version of the determinant, but this impression is misleading. In some important respects the permanent is much less tractable than the determinant. Nonetheless, permanents have found a wide range of applications from pure combinatorics (e.g., counting problems involving permutations) right through to applied science (e.g., modeling subatomic particles). For further reading, see [Min78], [Min83], [Min87], [CW05], and the references therein.

43.1 Basic Concepts

Definitions:

Let A = [aij] be an m × n matrix over a commutative ring, m ≤ n. Let S be the set of ...