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 ±α–permanent of a positive semi-definite Hermitian n×n matrix and the α/2–permanent of a positive semi-definite real symmetric n× n matrix if α is a nonnegative integer or α ≥ n−1. We are unable to settle Shirai’s nonnegativity conjecture for α–permanents when α ≥ 1, but we verify it up to the 5 × 5 case, in addition to recovering and refining some of Shirai’s partial results by purely combinatorial proofs.