  The aim of this paper is to present new upper bounds for the distance between
a properly normalized permanent of a rectangular complex matrix and the product
of the arithmetic means of the entries of its columns. It turns out that the
bounds improve on those from earlier work. Our proofs are based on some new
identities for the above-mentioned differences and also for related expressions
for matrices over a rational associative commutative unital algebra. Some of
our identities are generalizations of results in Dougall (Proceedings of the
Edinburgh Mathematical Society, 24, 61-77, 1905). Second order results are also
included.
