A Determinantal Identity for the Permanent of a Rank 2 Matrix

Abstract

We prove an identity relating the permanent of a rank 2 matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity of Carlitz and Levine, suggesting the possibility that these are actually special cases of some more general identity (or class of identities) connecting permanents and determinants. The proof combines some basic facts from the theory of symmetric functions with an application of a famous theorem of Binet and Cauchy in linear algebra.

MSC::

• Primary 15A15
• Secondary 05E05

Acknowledgments

The author wishes to recognize the support of the National Science Foundation (NSF CAREER grant DMS-1552520) as well as the input of two anonymous referees.

Additional information

Funding

The author wishes to recognize the support of the National Science Foundation (NSF CAREER grant DMS-1552520) as well as the input of two anonymous referees.

Notes on contributors

Adam W. Marcus

ADAM W. MARCUS received his Ph.D. in the ACO program at the Georgia Institute of Technology. He held positions at Yale and Princeton and spent three years as Chief Scientist at a startup before joining EPFL as the Chair of Combinatorial Analysis. He has a wide range of interests, but high among them are polynomials, linear algebra, and using the former to solve problems related to the latter.