Remarks on the $α$--permanent

We recall Vere-Jones's definition of the $α$--permanent and describe the connection between the (1/2)--permanent and the hafnian. We establish expansion formulae for the $α$--permanent in terms of partitions of the index set, and we use these to prove Lieb-type inequalities for the $\pmα$--permanent of a positive semi-definite Hermitian $n\times n$ matrix and the $α/2$--permanent of a positive semi-definite real symmetric $n\times n$ matrix if $α$ is a nonnegative integer or $α\ge n-1$. We are unable to settle Shirai's nonnegativity conjecture for $α$--permanents when $α\ge 1$, but we verify it up to the $5\times 5$ case, in addition to recovering and refining some of Shirai's partial results by purely combinatorial proofs.

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