Permanents through probability distributions
Abstract
We show that the permanent of a matrix can be written as the expectation value of a function of random variables each with zero mean and unit variance. This result is used to show that Glynn's theorem and a simplified MacMahon theorem extend from a common probabilistic interpretation of the permanent. Combining the methods in these two proofs, we prove a new result that relates the permanent of a matrix to the expectation value of a product of hyperbolic trigonometric functions, or, equivalently, the partition function of a spin system. We conclude by discussing how the main theorem can be generalized and how the techniques used to prove it can be applied to more general problems in combinatorics.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.11861
 arXiv:
 arXiv:2106.11861
 Bibcode:
 2021arXiv210611861W
 Keywords:

 Mathematics  Combinatorics;
 Mathematical Physics;
 15A15;
 68Q87;
 G.2.1
 EPrint:
 8 pages, 1 figure