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

A Determinantal Identity for the Permanent of a Rank 2 Matrix