Remarks on the $\alpha$--permanent

We recall Vere-Jones’s definition of the $\alpha$–permanent and describe the connection between the (1/2)–permanent and the hafnian. We establish expansion formulae for the $\alpha$–permanent in terms of partitions of the index set, and we use these to prove Lieb-type inequalities for the $\pm \alpha$–permanent of a positive semi-definite Hermitian $n\times n$ matrix and the $\alpha \mathrm{/}2$–permanent of a positive semi-definite real symmetric $n\times n$ matrix if $\alpha$ is a nonnegative integer or $\alpha \ge n-1$. We are unable to settle Shirai’s nonnegativity conjecture for $\alpha$–permanents when $\alpha \ge 1$, 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.